Richard Dedekind Theory of Algebraic Integers

THEORY OF ALGEBRAIC INTEGERS

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Theory of Algebraic Integers

Richard Dedekind

Translated and introduced by John Stillwell

CAMBRIDGEUNIVERSITY PRESS

Published by the Press Syndicate of the University of CambridgeThe Pitt Building, Trumpington Street, Cambridge CB2 1RP

40 West 20th Street, New York, NY 10011-4211, USA10 Stamford Road, Oakleigh, Melbourne 3166, Australia

First published in French 1877

English translation and introduction© Cambridge University Press 1996

First published in English 1996

Library of Congress cataloging in publication dataDedekind, Richard, 1831-1916.

Theory of algebraic integers / Richard Dedekind;translated and with an introduction by John Stillwell

p. cm.Includes bibliographical references and index.

ISBN 0-521-56518-9 (pbk.)1. Algebraic number theory. 2. Integral representations.

1. TitleQA247.D43 1996

512'.74-dc2O 96-1601 CIP

British Library cataloguing in publication data available

ISBN 0 521 56518 9 paperback

Transferred to digital printing 2004

Contents

Part one: Translator's introduction page 1T'ranslator's introduction 30.1 General remarks 3

0.2 Squares 60.2.1 Pythagorean triples 6

0.2.2 Divisors and prime factorisation 70.2.3 Irrational numbers 8

0.2.4 Diophantus 80.3 Quadratic forms 10

0.3.1 Fermat 10

0.3.2 The grit in the oyster 12

0.3.3 Reduction of forms 13

0.3.4 Lagrange's proof of the two squares theorem 150.3.5 Primitive roots and quadratic residues 16

0.3.6 Composition of forms 170.3.7 The class group 19

0.4 Quadratic integers 210.4.1 The need for generalised "integers" 21

0.4.2 Gaussian integers 220.4.3 Gaussian primes 240.4.4 Imaginary quadratic integers 25

0.4.5 The failure of unique prime factorisation 270.5 Roots of unity 290.5.1 Fermat's last theorem 290.5.2 The cyclotomic integers 300.5.3 Cyclotomic integers and quadratic integers 320.5.4 Quadratic reciprocity 360.5.5 Other reciprocity laws 38

v

vi Contents

0.6 Algebraic integers 390.6.1 Definition 390.6.2 Basic properties 400.6.3 Class numbers 41

0.6.4 Ideal numbers and ideals 420.7 The reception of ideal theory 440.7.1 How the memoir came to be written 440.7.2 Later development of ideal theory 45Acknowledgements 47Bibliography 48

Part two: Theory of algebraic integers 51

Introduction 53

1 Auxiliary theorems from the theory of modules 62§1. Modules and their divisibility 62§2. Congruences and classes of numbers 64§3. Finitely generated modules 67§4. Irreducible systems 71

2 Germ of the theory of ideals 83§5. The rational integers 83§6. The complex integers of Gauss 84§7. The domain o of numbers x + y/ 86§8. Role of the number 2 in the domain o 89§9. Role of the numbers 3 and 7 in the domain o 91

§10. Laws of divisibility in the domain o 93§11. Ideals in the domain o 95§12. Divisibility and multiplication of ideals in o 983 General properties of algebraic integers 103§13. The domain of all algebraic integers 103§14. Divisibility of integers 105

§15. Fields of finite degree 106§16. Conjugate fields 108§17. Norms and discriminants 111

§18. The integers in a field Il of finite degree 113

4 Elements of the theory of ideals 119§19. Ideals and their divisibility 119§20. Norms 121

§21. Prime ideals 123§22. Multiplication of ideals 125

§23. The difficulty in the theory 126§24. Auxiliary propositions 128

Contents vii

§25. Laws of divisibility 129

§26. Congruences 134

§27. Examples borrowed from circle division 138

§28. Classes of ideals 146

§29. The number of classes of ideals 147

§30. Conclusion 149

Index 153

Part oneTranslator's introduction

Translator's introduction

0.1 General remarks

Dedekind's invention of ideals in the 1870s was a major turning point inthe development of algebra. His aim was to apply ideals to number the-ory, but to do this he had to build the whole framework of commutativealgebra: fields, rings, modules and vector spaces. These concepts, to-gether with groups, were to form the core of the future abstract algebra.At the same time, he created algebraic number theory, which becamethe temporary home of algebra while its core concepts were growingup. Algebra finally became independent in the 1920s, when fields, ringsand modules were generalised beyond the realm of numbers by EmmyNoether and Emil Artin. But even then, Emmy Noether used to say"Es steht schon bei Dedekind" ("It's already in Dedekind"), and urgedher students to read all of Dedekind's works in ideal theory.

Today this is still worthwhile, but not so easy. Dedekind wrote foran audience that knew number theory - especially quadratic forms -but not the concepts of ring, field or module. Today's readers probablyhave the opposite qualifications, and of course most are not fluent inGerman and French. In an attempt to overcome these problems, I havetranslated the most accessible of Dedekind's works on ideal theory, Surla Theorie des Nombres Entiers Algebriques, Dedekind (1877), which hewrote to explain his ideas to a general mathematical audience. Thismemoir shows the need for ideals in a very concrete case, the numbersm + n/ where m, n E Z, before going on to develop a general theoryand to prove the theorem on unique factorisation into prime ideals.

The algebraic integers in Dedekind's title are a generalisation of theordinary integers - created in response to certain limitations of classicalnumber theory. The ordinary integers have been studied since ancient

3

4 Translator's introduction

times, and their basic theory was laid down in Euclid's Elements (seeHeath (1925)) around 300 BC. Yet even ancient number theory containsproblems not solvable by Euclid's methods. Sometimes it is necessaryto use irrational numbers, such as v,"2-, to answer questions about theordinary integers. A famous example is the so-called Pell equation

x2-cy2=1

where c is a nonsquare integer and the solutions x, y are required to beintegers. Solutions for certain values of c were known to the ancients, butthe complete solution was not obtained until Lagrange (1768) related theequation to the continued fraction expansion of He also showed thateach solution is obtained from a certain "minimum" solution (xo, yo) bythe formula

xk+ykf -±(xo+yo.)k.The irrational numbers xk + in this formula are examples of alge-braic integers, which are defined in general to be roots of equations ofthe form

an_ 1, ... , ao are ordinary integers, that is, an_ 1 i ... , ao E Z.Algebraic integers are so called because they share some properties

with the ordinary integers. In particular, they are closed under sum,difference and product, and the rational algebraic integers are just theordinary integers (for more details, see 0.6.2 and §13 of Dedekind's mem-oir). Because of the second fact, the ordinary integers are also knownas rational integers. The first fact implies that the algebraic integersform a ring. However, we are not interested in the ring of all algebraicintegers so much as rings like

Z[v]={x+yV':x,yEZ}and

Z[i]={x+yi:x,yeZ}.

In general we use the notation Z[a] to denote the closure of the setZ U {a} under +, - and x. The reason for working in rings Z[a] isthat they more closely resemble Z, and hence are more likely to yieldinformation about Z.

Any ring R of algebraic integers includes Z, so theorems about Zmay be obtainable as special cases of theorems about R (we shall see

0.1 General remarks 5

several examples later). However, useful theorems about R are provableonly when R has all the basic properties of Z, in particular, uniqueprime factorisation. This is not always the case. Z[-5] is the simplestexample where unique prime factorisation fails, and this is why Dedekindstudies it in detail. His aim is to recapture unique prime factorisation byextending the concept of integer still further, to certain sets of algebraicintegers he calls ideals. This works only if the size of R is limited in someway. The ring A of all algebraic integers is "too big" because it includesf along with each algebraic integer a. This gives the factorisationa = \I-a-vra- and hence "primes" do not exist in A, let alone uniqueprime factorisation.

Dedekind found the appropriate "small" rings R in algebraic numberfields of finite degree, each of which has the form Q(a), where a is analgebraic integer. Q(a) denotes the closure of Q U {a} under +,-,xand = (by a nonzero number), and each Q(a) has its own integers,which factorise into primes. In particular, Z[\] is the ring of integersof Q(/ ), and 6 = 2 x 3 is a prime factorisation of 6. Not the primefactorisation, alas, because 6 = (1+/)(1-) is also a factorisationinto primes (see 0.4.5). However, unique prime factorisation is regainedwhen one passes to the ideals of Z[v/-5], and Dedekind generalises thisto any Q(a). The result is at last a theory of algebraic integers capableof yielding information about ordinary integers.

A lot of machinery is needed to build this theory, but Dedekind ex-plains it well. Suffice to say that fields, rings and modules arise verynaturally as sets of numbers closed under the basic operations of arith-metic. Fields are closed under +, -, x and =, rings are closed under +,- and x, while modules are closed under + and -. The term "ring" wasactually introduced by Hilbert (1897); Dedekind calls them "domains"here, and I have thought it appropriate to retain this terminology, sincethese particular rings are prototypes of what are now called Dedekinddomains. Dedekind presumably chose the name "module" because amodule M is something for which "congruence modulo M" is meaning-ful. His name for field, Korper (which also means "body" in German),was chosen to describe "a system with a certain completeness, fullnessand self-containedness; a naturally unified, organic whole", as he ex-plained in his final exposition of ideal theory, Dedekind (1894), §160.

What Dedekind does not explain is where Z[/] comes from, andwhy it is important in number theory. This is understandable, becausehis first version of ideal theory was a supplement to Dirichlet's numbertheory lectures, Vorlesungen caber Zahlentheorie (Dirichlet (1871)). In

6 0.2 Squares

the present memoir he also refers to the Vorlesungen frequently, so hisoriginal audience was assumed to have a good background in numbertheory, and particularly the theory of quadratic forms. Such a back-ground is less common today, but is easy and fun to acquire. Evenexperts may be surprised to learn how far back the story goes. Thespecific role of can be traced back to the anomalous behaviour ofthe quadratic form x2+5y2, first noticed by Fermat, and later explainedin different ways by Lagrange, Gauss and Kummer. But the reason forFermat's interest in x2 +5 y2 goes back much further, perhaps to theprehistory of mathematics in ancient Babylon. Let us begin there.

0.2 Squares0.2.1 Pythagorean triples

Integers a, b, c such that

a2 + b2 = C2

are one of the oldest treasures of mathematics. Such numbers occur asthe sides of right-angled triangles, and they may even have been used toconstruct right angles in ancient times. They are called Pythagoreantriples after Pythagoras, but they were actually discovered indepen-dently in several different cultures. The Babylonians were fascinatedby them as early as 1800 BC, when they recorded fifteen of them on atablet now known as Plimpton 322 (see Neugebauer and Sachs (1945)).Pythagorean triples other than the simplest ones (3,4,5), (5,12,13) or(8,15,17) are not easily found by trial and error, so the Babyloniansprobably knew a general formula such as

a=2uv, b=u2-v2, c=u2+v2which yields an unlimited supply of Pythagorean triples by substitutingdifferent integers for u, v.

The general solution of a2 + b2 = c2 is in fact

a = 2uvw, b = (u2 - v2)w, c = (u2 + v2)w,

as may be found in Euclid's Elements Book X (lemma after Proposition29). A key statement in Euclid's proof is: if the product of relativelyprime integers is a square, then the integers themselves are squares.

Euclid first used the general formula for Pythagorean triples in his the-ory of irrational numbers, and it is in a different book from his theoryof integers. The assumption that relatively prime integers are squares

0.2.2 Divisors and prime factorisation 7

when their product is a square is justified by a long chain of proposi-tions, stretching over several books of the Elements. However, a directjustification is possible from his theory of integer divisibility, which is inBook VII. This theory is fundamental to the theory of ordinary integers,and also the inspiration for Dedekind's theory of ideals, so we should re-call its main features before going any further. Among other things, itidentifies the important but elusive role of primes.

0.2.2 Divisors and prime factorisationAn integer m divides an integer n if n = ml for some integer 1. We alsosay that m is a divisor of n, or that n is a multiple of m. An integerp whose only divisors are ±1 and ±p is called a prime, and any integercan be factorised into a finite number of primes by successively findingdivisors unequal to ±1 but of minimal absolute value. However, it is notobvious that each factorisation of an integer n involves the same set ofprimes. There is conceivably a factorisation of some integer

n = p1p2...pi = g1g2...qj

into primes p1, P2, , pz and q1, q 2 ,... , qj respectively, where one of theprimes p is different from all the primes q.

Nonunique prime factorisation is ruled out by the following proposi-tion of Euclid (Elements, Book VII, Proposition 30).

Prime divisor property. If p is prime and p divides the product ab ofintegers a, b, then p divides a or p divides b.

An interesting aspect of the proof is its reliance on the concept ofgreatest common divisor (gcd), particularly the fact that

gcd(a, b) = ua + vb for some integers u, v.

The set {ua + vb : u, v E 7L} is in fact an ideal, and unique prime factori-sation is equivalent to the fact that this ideal consists of the multiplesof one of its members, namely gcd(a, b).

It should be mentioned that Euclid proves only the prime divisor prop-erty, not unique prime factorisation. In fact its first explicit statementand proof are in Gauss (1801), the Disquisitiones Arithmeticae, article16. As we shall see, this is possibly because Gauss was first to recognisegeneralisations of the integers for which unique prime factorisation isnot valid.

8 0.2 Squares

0.2.3 Irrational numbersAs everybody knows, Pythagorean triples also have significance as thesides of right-angled triangles. In any right-angled triangle, the sidelengths a, b, c satisfy

a2 + b2 = c2

whether or not a, b and c are integers (Pythagoras' theorem). Hence itis tempting to try to interpret a right-angled triangle as a Pythagoreantriple by choosing the unit of length so that a, b and c all become integerlengths. Pythagoras or one of his followers made the historic discoverythat this is not always possible. The simplest counterexample is thetriangle with sides 1, 1, v'2-. It is impossible to interpret this triangle asa Pythagorean triple because is not a rational number.

A proof of this fact, which also proves the irrationality of i, f ,f and so on, uses unique prime factorisation to see that each primeappears to an even power in a square. Then the equation

2n2 = m2

is impossible because the prime 2 occurs an odd number of times in theprime factorisation of 2n2, and an even number of times in the primefactorisation of m2.

The irrationality of led the Greeks to study the so-called Pellequation

x2-2y2=1.

They found it could used to approach f rationally, via increasinglylarge integer solutions, xn, yn. Since xn - 2yn = 1, the quotient xn/ynnecessarily tends to v f2-. The general Pell equation

x2 - cy2 = 1, where c is a nonsquare integer,

can similarly be used to approach the irrational number V /c-. This equa-tion later proved fruitful in many other ways; Dedekind even used it toprove the irrationality of VFc (Dedekind (1872), Section IV).

0.2..4 DiophantusThe equations a2+b2 = c2 and x2-2y2 = 1 are examples of what we nowcall Diophantine equations, after Diophantus of Alexandria. Diophantuslived sometime between 150 AD and 350 AD and wrote a collection ofbooks on number theory known as the Arithmetica (Heath (1910)). They

0. 2.4 Diophantus 9

consist entirely of equations and ingenious particular solutions. Theterm "Diophantine" refers to the type of solution sought: either rationalor integer. For Diophantus it is usually a rational solution, but for someequations, such as the Pell equation, the integer solutions are of moreinterest. The Pell equation was actually not studied by Diophantus, buthe mentioned an integer solution to another remarkable equation: thesolution x = 5, y = 3 of y3 = x2 + 2. (See 0.4.1 for the astonishingsequel to this solution.)

Although all Diophantus' solutions are special cases, they usually seemchosen to illustrate general methods. Euler (1756) went so far as to say

Nevertheless, the actual methods that he uses for solving any of his problemsare as general as those in use today ... there is hardly any method yet inventedin this kind of analysis not already traceable to Diophantus. (Euler OperaOmnia 1,2, p. 429-430.)

And if anyone would know, Euler would. The first mathematician tounderstand Diophantus properly was Fermat (1601-1665), but his com-ments were as cryptic as the Arithmetica itself. Euler spent about 40years, off and on, reading between the lines of Fermat and Diophantus,until he could reconstruct most their methods and prove their theorems.We shall study the connection between Diophantus, Fermat and Eulermore thoroughly later, but one example is worth mentioning here. Itshows how much theory can be latent in a single numerical fact.

In the Arithmetica, Book III, Problem 19, Diophantus remarks

65 is naturally divided into two squares in two ways, namely into 72 +4 2 and82 + 12, which is due to the fact that 65 is the product of 13 and 5, each ofwhich is the sum of two squares.

It appears from this that Diophantus is aware of the identity

(a2 + b2)(c2 + d2) = (ac ± bd)2 + (ad bc) 2

though he makes no such general statement. However, Fermat saw muchdeeper than this. Noticing, with Diophantus, that the identity reducesthe representations of a number as a sum of two squares to the repre-sentations of its prime factors, his comment on the problem is:

A prime number of the form 4n + 1 is the hypotenuse of a right-angled triangle[that is, a sum of squares] in one way only ... If a prime number which is thesum of two squares be multiplied by another prime number which is also thesum of two squares, the product will be the sum of two squares in two ways.(Heath (1910), p. 268.)

10 0.3 Quadratic forms

The restriction to primes of the form 4n + 1 is understandable becausea prime p 54 2 cannot be the sum of two squares unless it is of the form4n+ 1 (by a congruence mod 4 argument). But Fermat's claim that anyprime p = 4n + 1 is a sum of two squares comes right out of the blue.No one knows how he proved it and the first known proof is due to Euler(1756). As we shall see later, Lagrange, Gauss and Dedekind all usedthis theorem of Fermat to test the strength of new methods in numbertheory.

0.3 Quadratic forms0.3.1 Fermat

Unlike Euclid or Diophantus, Fermat never wrote a book. His reputa-tion rests on a short manuscript containing his discovery of coordinategeometry (independent of Descartes), his letters, and his marginal noteson Diophantus. He took up number theory only in his late 30s, andleft only one reasonably complete proof, in the posthumously publishedFermat (1670). However, it is a beautiful piece of work, and fully estab-lishes his credentials as both an innovator and a student of the ancients.It also has a place in our story, as an application of Pythagorean triples,and as the first proven instance of Fermat's last theorem. Fermat's proofshows that there are no positive integers x, y, z such that x4 + y4 = z4,by showing that there are not even positive integers x, y, z such thatx4 + y4 = z2. It turns out to be the only instance of Fermat's last the-orem with a really elementary proof, involving just Euclid's theory ofdivisibility.

The argument is by contradiction, and the gist of it is as follows (omit-ting mainly routine checks that certain integers are relatively prime).

Suppose that there are positive integers x, y, z such that x4+y4 = z2,or in other words, (x2)2 + (y2)2 = z2

. This says that x2, y2, z is aPythagorean triple, which we can take to be primitive, hence there areintegers u, v such that

x2 = 2uv, y2 = u2 - v2, z = u2 + v2,

by Euclid's formulas (0.2.1). The middle equation says that v, y, uis also a Pythagorean triple, and it is also primitive, hence there areintegers s, t such that

v = 2st, y = s2 - t2, u = s2 + t2.

0.3.1 Fermat 11

This gives

x2 = 2uv = 4st(s2 + t2),

so the relatively prime integers s, t and s2 + t2 have product equal tothe square (x/2)2. It follows that each is itself a square, say

2 2, t=yi, s2+t2=zi,s=xi

and hence

4 4= 2x1 +y 1 - zl.

Thus we have found another sum of two fourth powers equal to a square,and by retracing the argument we find that the new square z2 is smallerthan the old, z2, but still nonzero. By repeating the process we cantherefore obtain an infinite descending sequence of positive integers,which is a contradiction.

Fermat called the method used in this proof infinite descent, and usedit for many of his other theorems. He claimed, for example, to haveproved that any prime of the form 4n + 1 is a sum of two squares bysupposing p = 4n + 1 to be a prime not the sum of two squares, andfinding a smaller prime with the same property. However, it is very hardto see how to make the descent in this case. Euler (1749) found a proofonly after several years of effort. In 0.3.4 we shall see an easier proofof the two squares theorem due to Lagrange. Lagrange's proof doesuse another famous theorem of Fermat, but it is the easy one knownas Fermat's "little" theorem: for any prime number p, and any integera 0- 0 (mod p), we have aTi-1 - 1 (mod p) (Fermat (1640b)).

The proof of Fermat's little theorem most likely used by Fermat usesinduction on a and the fact that a prime p divides each of the binomialcoefficients

p-

p(p - 1)(p - 2) (p - i + 1)for 1 < i < p - 1

i i!,- -

as is clear from the fact that p is a factor of the numerator but not ofthe denominator. This proof implicitly contains the "mod p binomialtheorem",

(a + b) P - a1' + b'' (mod p),

which has its uses elsewhere (see for example Gauss's proof of quadraticreciprocity in 0.5.4).

The proof more often seen today is based on that of Euler (1761),


which implicitly uses the group properties of multiplication mod p, par-ticularly the idea of multiplicative inverses. An integer a is nonzeromod p if gcd(a, p) = 1, in which case 1 = ar + ps for some integersr, s by the Euclidean algorithm. The number r is called a multi-plicative inverse of a (mod p) since ar - 1 (mod p). It follows thatmod p multiplication by a nonzero a is invertible, and in particular theset {a x 1, a x 2, ... , a x (p - 1)} is the same set (mod p) as the set{ 1, 2, ... , p - 11. Hence each set has the same product mod p,

ax2ax...x(p-1)a-lx2x...x(p-1) (modp),and cancellation of 1, 2, ... , p - 1 from both sides (which is permissible,since 1, 2, ... , p - 1 have inverses) gives Fermat's little theorem:

aP-1 - 1 (mod p).

0.3.2 The grit in the oysterThe mathematical pearl that is Dedekind's theory of ideals grew in re-sponse to a tiny irritant, the anomalous behaviour of the quadratic formx2 +5 y2 . Between 1640 and 1654 Fermat discovered three beautiful the-orems about the representation of odd primes p by the forms x2 + y2,x2 +2 y2 and x2 +3 y2 for integer values of x and y (the first promptedby Diophantus' remark on sums of squares, as mentioned in 0.2.4):

Theorem 1. p = x2 + y2 p = 1 (mod 4) (Fermat (1640c))

Theorem 2. p = x2 + 2y2 p 1 or 3 (mod 8) (Fermat (1654))

Theorem 3. p = x2 +3 y2 p 1 (mod 3) (Fermat (1654))

Fermat thought he could prove these theorems, and he was probablyright, as proofs eventually published by Euler were based on Fermat'smethod of infinite descent. Since x2 +4 y2 = x2 + (2y)2, which is "ofthe form" x2 + y2, the next theorem should be about x2 +5 y2 . Thisis the grit in the oyster. Fermat was unable to prove a theorem aboutprimes of the form x2 +5 y2 , and could only conjecture the following lesssatisfying fact.

If two primes which end in 3 or 7 and surpass by 3 a multiple of 4 are multiplied,then their product will be composed of a square and the quintuple of anothersquare. (Fermat (1654).)

Since numbers that end in 3 or 7 are of the form 10n + 3 or lOn + 7, and

0.3.3 Reduction of forms 13

such a number is also of the form 4m + 3 if and only if it is of the form20k + 3 or 20k + 7, Fermat's conjecture can be restated as follows:

Fermat's conjecture. If two primes are of the form 20k + 3 or 20k + 7then their product is of the form x2 + 5y2.

This is very puzzling. What about primes of the form x2 + 5y2 ?There are some, such as

29=32+5x22,41=62+5x12,61=42+5x32,

and furthermore, they lie in the classes 20k + 1 and 20k + 9 that seemconspicuously absent from the conjecture above. This situation begs foran explanation. As Weil (1974) says:

When there is something that is really puzzling and cannot be understood,it usually deserves the closest attention because some time or other some bigtheory will emerge from it.

Euler (1744) found another clue to the puzzle. He noticed two "faces" tothe behaviour of x2 + 5y2 - sometimes it represents a prime p, sometimes2p - and he conjectured that the following holds for all prime values p.

Euler's conjecture. p = x2 +5y2 p m 1 or 9 (mod 20)2p = x2 +5 y2 p m 3 or 7 (mod 20).

0.3.3 Reduction of formsThe first to account rigorously for the two-faced behaviour of x2+5y2 wasLagrange (1773). Studying the general question of which integers n couldbe represented by a given quadratic form axe+bxy+cy2 (where a, b, c EZ), he had the very fruitful idea of finding those forms a'x'2+b'x'y'+c'y'2equivalent to ax2 + bxy + cy2 via a change of variables.

The substitution

x'=ax+/3y,'y =ryx+Sy

maps Z x Z into Z x Z provided a, 0,,y, 6 E Z and it is one-to-one providedthere is an inverse substitution

x = a'x' + /3'y',

y=7x +6'y'.


In this case

[yS)[ry'

s-]-[0 1

since the product of a substitution and its inverse is the identity. There-fore, taking determinants of both sides,

a

y S

a' /3'

y' S'=1.

Finally, since the determinants on the left are integers and the onlyinteger divisors of 1 are ±1, it follows that the invertible substitutionsfor which the pairs (x', y') run through Z x Z are precisely those witha, 0, y, S E Z and determinant aS - Qy = f1. Such substitutions arenow called unimodular.

The result a'xj2+b'x'y'+c'y'2 of a unimodular substitution a'x'+/3'y'for x and y'x' + S'y' for y in ax 2 + bxy + cy2 is therefore a form thattakes the same values as ax 2 + bxy + cy2. Forms transformable intoeach other by unimodular substitutions are equivalent, as we would say,because the unimodular substitutions form a group. Lagrange observedthat equivalent forms have the same discriminant

D=b2-4ac=b'2-4a'c',as can be checked by computing bj2 - 4a'c' and using aS - 0y = ±1.(Incidentally, the old term for the discriminant of a quadratic form wasdeterminant. I have retained this term in the translation of Dedekind'smemoir because he refers to a slightly different definition, due to Gauss.)Observing the invariance of the discriminant is a first step towards decid-ing equivalence of forms. To go further we need to answer the question:how many inequivalent forms have the same discriminant?

Lagrange found a way to answer this question for forms with negativediscriminant. He showed that any ax 2 + bxy + cy2 can be transformedinto an equivalent form a'x'2 + b'x'y' + c'y'2 that is reduced in the sensethat Ib'I < a' < c'.

It follows that

-D=4a'c'-bj2>4aj2-a'2=3a'2and therefore, in the case of negative discriminant, only finitely manyvalues of the integers a', b' and hence c', can occur in reduced forms.For any particular D < 0 it is then possible to work out the inequivalentreduced forms of discriminant D. The number of them is called the classnumber h(D). The first few calculations yield the following results.

0.3.4 Lagrange's proof of the two squares theorem 15

All forms with discriminant -4 are equivalent to x2 + y2, henceh(-4) = 1.

All forms with discriminant -8 are equivalent to x2 +2 y2 , henceh(-8) = 1.

All forms with discriminant -12 are equivalent to x2 +3 y2 , henceh(-12) = 1.

But there are two inequivalent reduced forms with discriminant -20,the forms x2 +5 y2 and 2x2 + 2xy + 3y2, so h(-20) = 2. Aha! Thereis something different about x2 +5 y2f Perhaps the previously invisiblecompanion 2x2+2xy+3y2 accounts for its "two-faced" behaviour. Beforefollowing up this suggestion, however, let us see how Lagrange used theuniqueness of the form x2 + y2 to prove the two squares theorem.

0.3.4 Lagrange's proof of the two squares theoremGiven a prime p = 4n + 1, it suffices to find any form with discriminant-4 that represents p, since Lagrange has shown we can transform thisform into x2 + y2. It suffices in turn to find an integer m such thatp divides m2 + 1, because in this case the form px2 + 2mxy + M2+1 y2

has integer coefficients, its discriminant is -4, and of course it takesthe value p for x = 1, y = 0. (The use of this particular form is asimplification of Lagrange's argument due to Gauss (1801), article 182).

This is where it is crucial that p be of the form 4n + 1 and prime. ByFermat's little theorem, p then divides z4n - 1 = (z2n - 1)(z2n + 1) forany integer z relatively prime to p. Thus p will divide z2n + 1, whichis m2 + 1 with m = zn, provided z is chosen so that p does not dividez2n - 1. In terms of congruences, we want to choose one of the 4nnonzero values of z (mod p) so that z2n - 1 is nonzero (mod p). Thisis possible by an earlier theorem of Lagrange (1770): a congruence ofdegree q modulo a prime p has at most q solutions.

A modern proof of this theorem (not much different from Lagrange'sown proof) uses the fact that the nonzero integers mod p have multi-plicative inverses. It follows that the congruence classes mod p form afield, and one can show as in classical algebra that each root zi of apolynomial p(z) corresponds to a factor (z - zi) of p(z). Hence therecannot be more roots than the degree of p(z).

It follows in the present case that we have at least 4n - 2n = 2nvalues of z which make z2n - 1 nonzero mod p, and hence p dividesz2n + 1 = m2 + 1 as required.


0.3.5 Primitive roots and quadratic residuesLagrange's result that an nth degree congruence mod p has at most n dif-ferent solutions has another important consequence in mod p arithmetic- the existence of primitive roots. We digress a little further here todiscuss this concept, since it will be important later, and it also throwsmore light on the two squares theorem.

An integer a is called a primitive root mod p if

aP-1 1 (mod p),

a' 1 (mod p) for l < q < p - 1.

More generally, if we define the order of element a mod p to be the leastpositive n such that an = 1 (mod p), then a primitive root is an elementof order p - 1. Existence of a primitive root means that the group ofnonzero congruence classes is cyclic. Euler conjectured that a primitiveroot exists for each p, but was unable to prove it. The first proof wasgiven by Gauss (1801), article 55. Like all proofs since, the primitive rootis not constructed explicitly - rather, its existence is shown by countingthe number of possible solutions of a congruence.

The quickest proof looks at elements of relatively prime order, andfirst shows that the least common multiple 1 of the orders of nonzerointegers mod p is itself the order of some element, a. Since 1 is the leastcommon multiple we then have

x1=-1 (modp) for x=1,2,...,p-1.If 1 < p - 1 this is a congruence of degree 1 with more than 1 solutions,contrary to Lagrange's result. Hence in fact 1 = p - 1, which means a isa primitive root mod p.

The existence of a primitive root a mod p means that

{l,a,a 2,...,aP-2} _ {1,2,...,p- 1}

This makes many facts about multiplication mod p quite transparent.F o r example, exactly h a l f the integers 1, 2, ... , p - 1 are squares, mod p.Indeed, the even powers 1, a2, a4, ... of a primitive root are squares modp, and conversely.

The traditional term for squares mod p is quadratic residues mod p.Quadratic residues arise naturally in the study of quadratic forms, andtheir fundamental theorem, called "quadratic reciprocity", was conjec-tured by Euler. Like the existence of primitive roots (also conjecturedby Euler), it was first proved by Gauss. We shall say more about this

0.3.6 Composition of forms 17

in Section 0.5. Two important preliminaries to the general discussion ofsquares mod p are the following:

Euler's criterion. m is a square mod p e* m 2 1 (mod p)

If a is a primitive root mod p,

m is square mod p = m=a2i for some jm221

= ai(P-1) = 1 (mod p),

m is nonsquare mod p #> m = a2j+1 for some jmp2 = ai(P-1)+p21 - ap21

-1 (mod p).

Quadratic character of -1. The number -1 is a square mod pp = 4n + 1 for some n.

Notice that this statement strengthens the result used in the proof ofthe two squares theorem, that p = 4n + 1 divides some m2 + 1. It alsofollows easily from the existence of primitive roots:

-1 is a square mod p x2 - -1 (mod p) for some x

x has order 4 (mod p)

e* x = a4 where a is a primitive roote#0. P = 4n + 1 for some n.

0.3.6 Composition of formsThe second reduced form of discriminant -20, namely 2x2 + 2xy + 3y2,accounts for much of the anomalous behaviour of x2 +5 y2

. The primesnot represented by x2 +5 y2 , namely those in the classes 20n + 3 and20n + 7, are represented by 2x2 + 2xy + 3y2, for example

3=2x02 +2x0x1+3x12,7=2x12+2x1x1+3x12,

23 = 2 x (-2)2 +2 x (-2) x 3+3 x 3 2.

Lagrange (1773) proved this, and also Euler's conjecture (0.3.2) that allprimes p in the classes 20n + 1 and 20n + 9 are of the form x2 +5 y2.He used these two theorems to establish the conjectures of Fermat andEuler (0.3.2) about products and doubles of primes in the classes 20n+3and 20n + 7, with the help of another observation: the product of two


numbers of the form 2x2 + 2xy + 3y2 is a number of the form x2 +5 y2.This crucial observation is based on the following algebraic identity:

(2x2 + 2xy + 3y2)(2x'2 + 2x'y' + 3yj2) = X2 + 5Y2

where

X = 2xx' + xy' + yx' - 2yy',Y=xy'+yx'+yy'.

Such an identity can be checked mechanically by multiplying out bothsides, but how did Lagrange find it in the first place? Probably by pastexperience with products of forms, some of which were known muchearlier. We have already seen one known to Diophantus (0.2.4):

(x2 + y2)(x 2 + y'2) = (xx - yy')2 + (xY1 + yx)2

It has a generalisation

(x2 + cy2)(x 2 + cy'2) = (xx' - cyy')2 + c(xy' + yx )2

discovered by the Indian mathematician Brahmagupta around 600 AD.(See Colebrooke (1817), p. 363, and Weil (1984), p. 14.)

This would have been easy for Lagrange to derive using the complexfactors

(x2+cy2)(x2+cyj2) =

and pairing the first factor with the third, and the second with thefourth. Lagrange's own identity can be derived quite mechanically from

(x2 + 5y2)(xj2 + 5yj2) = (xx' - 5yy')2 + 5(xy' + yx')2

(Brahmagupta's identity for c = 5), and

12x2+2xy+3y2 = 2 [(x+ 2)2+5(2)21

(the result of completing the square on 2x2 + 2xy + 3y2). Use the latterto rewrite each of 2x2 + 2xy + 3y2 and 2x'2 + 2x'y' + 3yj2, in the form2 [X2 + 5Y2}, multiply them out using Brahmagupta's identity, thenabsorb the factors of 2.

There is a related identity for the product of the two different formsof discriminant -20:

(x2 +5 Y2) (2x'2 + 2x'y' + 3 yj2) = 2X2 + 2XY + 3Y2

0.3.7 The class group 19

where

X = xx' - yx' - 3yy',Y = xy' + 2yx' + yy'

and it can be derived in a similar way. These identities show that theforms x2 +5 y2 and 2x2 + 2xy + 3y2 are "closed under products" in acertain sense. The product operation is known as composition of forms.

Legendre (1798) managed to show, in fact, that any two quadraticforms with the same discriminant could be "composed" in this fashion.Something very interesting was going on, but what?

0.3.7 The class groupComposition of forms came on the scene decades before the axiomaticproperties of abstract structures, such as groups, were considered inmathematics. Legendre had found a set (the forms with fixed negativediscriminant) and a "product" operation on it (composition) but therewas no reason to expect the operation to have simple or interestingstructural properties. All he could was draw up "multiplication tables"for the forms with particular discriminants. For example, if we take theforms with discriminant -20,

A = x2 +5y2 ,

B = 2x2 + 2xy + 3y2,

the table would be

AB

A BA BB A

because AA = A by Brahmagupta's identity, BB = A by Lagrange'sidentity, and AB = BA = B by the last identity of 0.3.6.

A more complicated example actually given by Legendre was for theforms of discriminant -164, rewritten by Cox (1989)) p. 42 as:

A = x2 + 41y2,

B = 2x2 + 2xy + 21y2,

C = 5x2 + 6xy + 10y2,

D = 3x2 + 2xy + 14y2,

E = 6x2 + 2xy + 7 y2.


Legendre's multiplication table for these forms can then be written:

A B C D EA A B C D EB B A C E DC C C AorB DorE DorED D E DorE AorC BorCE E D DorE BorC AorC

To us, of course, it is rather disturbing to see AC = BC when A # B,and worse still that some of the products have two values. The two-valued entries are due to an ambiguity of sign in Legendre's definitionof composition. Ambiguity would be avoided in any modern attempt todefine an operation on a set, and it was avoided in the next study ofquadratic forms, by Gauss (1801) (the Disquisitiones Arithmeticae).

Gauss's analysis of forms under composition is amazingly modern insome ways. He defined composition unambiguously, showed that it iswell-defined on equivalence classes of forms and showed, in effect, thatthe equivalence classes of forms constitute an abelian group under com-position. It is now called the class group. Gauss even came close tofinding a decomposition of the class group into cyclic factors.

Yet all this was accomplished with definitions and proofs so cumber-some it took 70 years for the rest of the world to understand them. WithGauss's definition of composition, for example, it is a major problem toprove that composition is associative. Even the statement of associativ-ity in the Disquisitiones is clumsy, as if Gauss had not really graspedwhat associativity is about:

If the form F is composed of the forms the form a from F and f"; theform F' from f, f"; the form 13' from F' and f'; then the forms %, a' will be... equivalent. (Gauss (1801), article 240.)

It is only by using the commutativity of composition (which is mercifullyobvious) that this statement can be rewritten in a form recognisable asassociativity, namely

f"(ff') = (f"f)f'.The proof is monstrous. It requires the derivation of 37 equations, mostof which Gauss leaves to the reader. This put the subject of compositionof forms out of the reach of most mathematicians until Dirichlet andDedekind simplified it enough to make associativity obvious. A polishedexposition was eventually given by Dedekind in his Supplement X to

0.4.1 The need for generalised "integers" 21

the 2nd edition of Dirichlet's Vorlesungen caber Zahlentheorie (Dirichlet(1871)).

However, the approach via algebraic identities, no matter how slick,was becoming irrelevant by this time. The abstract structure of the classgroup was seen as more important, and more usable, since facts could bededuced from it without reference to the definition of composition. Forexample, Kronecker (1870) showed that the decomposition of the classgroup into cyclic factors follows purely from axioms for finite abeliangroups (which he was the first to state). Dedekind's Supplement X toDirichlet (1871) was in effect the swansong of the old theory of compo-sition of forms, because in the same work he showed how to rebuild theclass group on a simpler and more general basis, the theory of algebraicintegers. The first important examples of such integers were studied byLagrange and Euler.

0.4 Quadratic integers0.4.1 The need for generalised "integers"

We have already seen how convenient it is to use factorisations involvingsquare roots to prove identities about integers, such as

(x2 + cy2)(x 2 + cy'2) = (xx - cyy')2 + c(xy' + yx)2.

Lagrange (1768) and Euler (1770) began using complex or irrationalfactorisations to find integer solutions of equations. The most interestingand provocative example is the Euler (1770) proof of the following claimof Fermat (1657): 27 is the only cube that exceeds a square by 2. Thisis equivalent to saying that y = 3, x = 5 is the only positive integersolution of the equation

y3=x2+2.

Euler's solution is breathtaking, even if not exactly rigorous. He fac-torises the right hand side into (x - /)(x + s), and proceeds totreat x - and x - as if they were integers. He assumes theyhave gcd 1, and therefore, since their product is a cube, they are cubesthemselves. This presumably means that

x + = (a + bV"--2)3 for some a, b E Z

= a3 + 3a2b/ + 3ab2(-2) + b3(-2)

22 0..4 Quadratic integers

and therefore

x=a3-6ab2,1 = 3a2b - 2b3

equating real and imaginary parts. But

1 = 3a2b - 20 = b(3a2 - b2)

only if b = ±1 and a = ±1, since 1 and -1 are the only integer divisorsof 1. This gives x = 5 as the only positive integer solution for x. Q.E.D!

Euler gave several examples of this kind, generally splitting quadraticsinto irrational complex factors and treating the factors as integers. Whyis this permissible, if indeed it is? To answer this question we need torecall how ordinary integers behave, particularly as divisors.

The behaviour of the ordinary integers is ruled by unique prime fac-torisation, which in turn depends on the prime divisor property: a primedivides a product ab only if it divides one of a, b (0.2.2). For example,this is crucial in proving that relatively prime integers are squares whentheir product is a square (0.2.1). To justify similar propositions about"quadratic integers" such as x + , we have to decide which of themare "primes" and then see whether they have a prime divisor propertylike the ordinary integers. This is easiest when the quadratic integers inquestion have a "Euclidean algorithm", because one can then follow thetrail blazed by Euclid in his proof of the prime divisor property.

The first to carry out such a program was Gauss (1832), who studiedthe divisibility properties of the numbers x + y , where x, y E Z.These numbers are now known as the Gaussian integers. They are thesimplest kind of quadratic integers, and they tie up nicely with quadraticforms and some other threads in our story, so it is worth looking at themfirst.

0.4.2 Gaussian integersMaking the usual abbreviation i for, we denote the set of Gaussianintegers x + yi by Z[i]. The first step towards unique prime factorisationin Z[i] is to show that primes exist. We can do this immediately withthe help of the norm, a measure of size in Z[i] introduced by Gauss. Thenorm of x + yi , N(x + yi), is defined by

N(x+yi) _ Ix+yi12 =x2+y2,

0.4.2 Gaussian integers

and has the multiplicative property

N((x + yi)(x' + y'i)) = N(x + yi)N(x' + y'i),

because

23

(xx' - yy')2 + (xy +yx )2 = (x2 +y2)(xj2 +Y,2),

by Diophantus' identity (0.2.4).It follows that any divisor x + yi ofX + Yi has N(x + yi) < N(X + Yi) and therefore, since the norm is apositive integer, each Gaussian integer of norm > 1 has a divisor x + yiof minimal norm > 1.

We call any such divisor a Gaussian prime because it is not the prod-uct of Gaussian integers of smaller norm. By repeatedly removing primedivisors, we obtain a factorisation of any Gaussian integer into Gaussianprimes. Obviously these factors are determined only up to factors ofnorm 1, the so-called units 1, -1, i, -i. This is similar to the situationin 7L, where prime factors are determined only up to the unit factors±1. The real problem in 7L[i], as in 7L, is to establish the prime divisorproperty: if a prime divides a product then it divides one of the factors.

Euclid's proof of the prime divisor property for Z (0.2.2) involves hisalgorithm for finding the gcd by repeated subtraction, which terminatesbecause it yields a decreasing sequence of positive integers. The proof forZ[i] is similar, but the algorithm for gcd requires repeated division withremainder, which terminates because it yields a sequence of remaindersthat decrease in norm. The decrease is guaranteed by the followingdivision property of Z[i]: for any Gaussian integers a and ,Q # 0 thereare Gaussian integers p and p ("quotient" and "remainder") such that

a=µ,d+p where 0 < 1pI < 1,31.

The division property is clear as soon as one realises that the multiplesp,3 of 3 lie at the corners of a lattice of squares in the complex plane. Atypical square is the one with corners 0, ,3, i,3, (1 +i)/3. The remainderp is the difference between a and the nearest corner p/3 in the lattice,so 0 < lpl < 1,31 because the distance IpI between any point in a squareand the nearest corner is less than the length 1,31 of a side.

To find gcd(a, 0) one can therefore let a = a1, ,3 = N1 and repeatedlycompute

aj+1 = ,aj,

Oj+i = remainder when aj is divided by (3

until a zero remainder /3k is obtained. (We use j, k.... for indices from

24 0.4 Quadratic integers

now on, since i will be reserved for .) Then gcd(a, /3) = 3k, andthere are Gaussian integers p and v such that gcd(a, /3) = pa + v/3 byan argument like that used for Z (0.2.2). In fact, the rest of the routeto unique prime factorisation is essentially the same as in Z. We get theprime divisor property by arguing that

1 = gcd(7r, a) = µ7r + va

when it is a Gaussian prime not dividing a, and multiplying both sidesby /3. Unique prime factorisation (up to unit factors) is obtained bysupposing

7172 ... 7rr = 0102 ... Os

are two prime factorisations of the same number, and cancelling 7r1, 7r2, .. .in turn until only units remain.

0.4.3 Gaussian primesIdentifying the actual primes of Z[i] is a separate question, and Gaussobserved that it is closely connected with Fermat's two squares theorem.This is because N(x + yi) = x2 + y2, so divisors of x + yi have normdividing x2 + y2. In particular, if x2 + y2 is an ordinary prime thenx + yi is a Gaussian prime, because it cannot be the product of numbersof smaller norm. Fermat's two squares theorem tells us that the primesof the form x2 + y2 are exactly the primes p = 4n + 1 and p = 2, so eachsuch x2 + y2 gives us two Gaussian primes, x + yi and x - yi. Moreover,the factorisation

x2 + y2 = (x + yi)(x - yi)

is unique up to unit factors, by unique prime factorisation in Z[i]. Hencethere are exactly two Gaussian primes, up to unit factors, for each or-dinary prime p = 4n + 1. (This shows, incidentally, that the partitionx2 + y2 of p into two squares is unique, a result also stated by Fermat.)The ordinary prime 2 is exceptional, being the square of the Gaussianprime 1 + i, up to a unit factor.

Conversely, if x + yi is a Gaussian prime then so is its conjugatex - yi, since any factorisation of x - yi yields a factorisation of x + yiby conjugation. Thus

x2 + y2 = (x + yi)(x - yi)

is a Gaussian prime factorisation, and hence unique up to unit factors.

0.4.4 Imaginary quadratic integers 25

It follows that x2 + y2 is an ordinary prime, since any factorisation of itinto ordinary primes would yield a different factorisation into Gaussianprimes.

Thus the "properly complex" Gaussian primes are (up to unit factors)the conjugate factors x + yi and x - yi of ordinary primes of the formx2+y2. By Fermat's two squares theorem these are the primes p = 4n+1and p = 2. The real Gaussian primes are the ordinary primes p = 4n+3,since these have no divisors of the form x2 + y2 (by Fermat's theoremagain) and hence no complex Gaussian prime divisors x + yi (since adivisor x + yi implies a divisor x - yi, by conjugation).

The close relation between Gaussian primes and the ordinary primesof the form x2 + y2 suggests that the theory of 7G[i] may be used to proveFermat's two squares theorem. Dedekind gave at least two such proofs,one in the memoir below (§27) and another in his final version of idealtheory (Dedekind (1894), §159). The latter proof is short enough todescribe here, since it builds on the result of Lagrange that p = 4n + 1divides a number of the form m2 + 1, which we have already describedin 0.3.4.

Since p does not divide either of the Gaussian factors m + i, m - iof m2 + 1 (the quotients p ± I are not Gaussian integers), p is nota Gaussian prime, by the prime divisor property of Z[i]. We thereforehave a Gaussian factorisation

p = (x + yi)(x - yi),

which gives

p=x2+y2.This argument from unique prime factorisation replaces Lagrange's ar-gument from the equivalence of quadratic forms with discriminant -4.In fact, Dedekind (1894), §159, goes on to deduce the equivalence ofthese forms directly from unique prime factorisation in Z[i].

0.4.4 Imaginary quadratic integersAs an obvious generalisation of the Gaussian integers we can consider,for any positive integer c,

7L[] = {x + yv/-c : x, y E 7L},

with norm defined by

N(x + yam) = Ix + yI2 = x2 + cy2.


The multiplicative property of the norm,

N((x + y\c)(x' + y'V)) = N(x + y )N(x' + y' ),

is equivalent to the Brahmagupta identity (0.3.6) and one proves, asin Z[i], that each element of Z[V/-c] has a factorisation into primes.However, the uniqueness of the prime factorisation depends on the valueof c, as we shall see in the next section.

After the Gaussian integers (c = 1), the case c = 2 is also of interest,since Z[v/-2] contains the numbers x± used by Euler in his solutionof the equation y3 = x2 + 2. Unique prime factorisation also holds inZ[y]. The proof, like that in Z[i], is based on the division property:for any a and /3 # 0 there are p and p in Z[vr-2] ("quotient" and"remainder") such that

a=p/3+p where 0<JpJ <1/31.

The division property holds in Z[\] for similar geometric reasons.The multiples p/3 of 0 lie at the corners of a lattice of rectangles in thecomplex plane. A typical rectangle is the one with corners 0, /3,,Q,(1 + / ),(3. The lengths of its sides are 1,31 and '1,31, so the maximumdistance of any point from a corner (attained by the centre point) is

a IQI < 1131, as required for the division property.

It follows that Z[\/-2] has a Euclidean algorithm, and hence uniqueprime factorisation, by the argument used for Z[i]. Since N(x+yv) =x2 +2 y2, the only units in Z['] are ±1, and prime factorisation isunique up to sign (as in Z, and hence "more unique" than in Z[i]). Theactual primes in Z[/] can be described using Fermat's theorem (0.3.2)that the (ordinary) primes of the form x2 + 2y2 are those of the formp = 8n + 1 or p = 8n + 3. Each ordinary prime p = 8n + 1 = x2 + 2y2or p = 8n + 3 = x2 +2y2 splits into primes x + y.,/----2 and x - yV---2of Z[V/-2], and each ordinary prime p = 8n + 5 or p = 8n + 7 remainsprime in Z[i].

Finally, to justify Euler's solution of y3 = x2 + 2 it remains to provethat

gcd(x+v,'--2,x-v)=1and that relatively prime numbers in Z[\] are cubes when their prod-uct is a cube.

To see why gcd(x + V/'--2, x-/) = 1, note that any common divisorof x + and x - is also a divisor of their difference 2 . Also,any solution x of y3 = x2 + 2 must be odd, because an even x yields

0.4.5 The failure of unique prime factorisation 27

an even y, in which case 8 divides y3 and does not divide x2 + 2. ThusN(x + / ) must be odd, and hence its gcd with N(2 v/ ) = 8 is 1.But this implies that 1 is the gcd of x - \ and 2 f , and hence ofx+ and x - .

Thanks to unique prime factorisation, and the fact that the units ofZ[v-2] are ±1, the proof that relatively prime numbers are cubes whentheir product is a cube is essentially the same as in Z. This in turn islike the proof for squares mentioned in 0.2.1.

0.4.5 The failure of unique prime factorisationIt should not be a great surprise that something goes wrong in 7G[],because the norm of x+y is the anomalous quadratic form x2+5y2.In short, prime factorisation is not unique. Consider the following twofactorisations of 6:

2x3=(1+V)(1-/).The norms of the factors are N(2) = 4, N(3) = 9, N(1 + V/--5) = 6and N(1 - V/'--5) = 6. The only nontrivial divisors of these normsare 2 and 3, which are not norms, hence the factors 2, 3, (1 + V1----5)1(1 - ) themselves have no nontrivial divisors. These factors aretherefore primes in Z[v/-5], and they certainly differ by more than unitfactors, since the norms of the factors on the left are different from thenorms of the factors on the right.

So here we have another anomalous behaviour of x2 + 5y2. Its ir-rational factors x + y and x - y/ belong to a set Z[/] of"integers" without unique prime factorisation. Can this be related tothe fact that x2 + 5y2 is one of two inequivalent quadratic forms withdiscriminant -20?

Indeed yes. The nonuniqueness of prime factorisation in Z[V/-5] canbe traced to the existence of ordinary integers x2 + 5y2 with divisorsnot of the same form. If each number x + y/ had a unique primefactorisation in Z[v/-5],

x+y/ = (x1+y1V-5)(x2+y2V)...(xr+Yr\),

then we should have the following factorisation of its norm in Z:

x2 + 5y2 = (x2 +5 Y2)(X2 + 5y2)...(xr + 5Y,2).

And each factor (x + 5y?) would be an ordinary prime or the squareof an ordinary prime, since any further decomposition of x2 + 5y2 into


ordinary primes would yield a different prime factorisation of x + yin Z[/]. Then, since the product of numbers of the form (x + 5y?)is another number of the same form (by Brahmagupta's identity), eachdivisor of x2+5y2 would also have this form. But of course 6 has divisors2 and 3, which are not of the form x2 + 5y2. Instead, they are of theother form with discriminant -20, namely 2x2 + 2xy + 3y2.

Most of this could have been noticed by Lagrange, but the failureof unique prime factorisation in Z[v/-5] was not explicitly pointed outuntil Kummer (1844) proposed the introduction of "ideal factors" to saveit. Kummer (1846b) went on to define a notion of equivalence for idealfactors, under which ideal factor classes correspond to classes of formswith the same discriminant. Thus inequivalence of forms is actually thesame thing as failure of unique prime factorisation. Kummer's discoverychanged the whole direction of number theory at this point, from thetheory of forms to the theory of ideal factors. However, it would be anoversimplification to say that this happened just because of quadraticforms like x2 + 5y2 . A second major influence was the theory of rootsof unity, which will be discussed in the next section.

It is possible that number theory could have changed direction muchearlier, if not for the conservatism of Gauss. According to Kummer(1846a), Gauss was aware of the failure of unique prime factorisation forquadratic integers when he wrote the Disquisitiones, and could see thatsomething like ideal factorisation was needed. However, he was unableto find a rigorous description of ideal factors, and invented the theoryof composition of forms as a substitute. He later told Dirichlet

If I wanted to proceed with the use of imaginaries in the way that earliermathematicians have done, then one of my earlier researches which is verydifficult [composition of forms] could have been done in a very simple way.(Excerpt from Kummer (1846a) in Edwards (1977), p. 335.)

Having invested so much energy in composition of forms, Gauss perhapsbecame unwilling to pursue the alternative theory of quadratic integers.As we know, it was only in 1832 (the paper on Gaussian integers) thathe got as far as proving unique prime factorisation for Z[i].

On the other hand, it must be admitted that Gauss's work on othertopics - the theory of regular polygons (or circle division) and the searchfor reciprocity laws - was crucial in the development of the more generalconcept of algebraic integer. As we shall now see, this work also belongsto the theory of roots of unity.

0.5.1 Fermat's last theorem 29

0.5 Roots of unity0.5.1 Fermat's last theorem

The Pythagorean equation is probably the most fruitful equation in num-ber theory, having been the inspiration for important results of Euclid(0.2.1), Diophantus (0.2.4) and Fermat (0.3.1). Diophantus threw newlight on the Pythagorean equation by considering rational solutions, in-stead of just integer solutions, and by showing that any nonzero squarein Q splits into two nonzero squares. Next to this result in Diophantus'Arithmetica, Fermat wrote his famous marginal note:

It is impossible to separate a cube into two cubes, or a biquadrate into twobiquadrates, or in general any power higher than second into powers of likedegree; I have discovered a truly marvellous proof of this which however thismargin is too small to contain.

This claim of Fermat's became known as Fermat's last theorem in theearly 19th century, as by then it was the only claim of Fermat's still tobe proved or disproved.

Fermat's last theorem is the claim that the equationan + bn = Cn

has no nonzero solution a, b, c E Q when n is an integer > 2. Since anyrational solution becomes an integer solution when multiplied throughby a common denominator, an equivalent claim is that an + bn = cnhas no nonzero integer solution. Although these two claims are logicallyequivalent, they suggest different viewpoints and different approachesto proving the the theorem. The approach through rational numbers,which stems from Diophantus, is "geometric" in a broad sense. Theapproach through integers, which stems from Fermat and Euler, is "al-gebraic". The geometric approach seems to have won the day, but onlyby calling on large amounts of algebra, analysis and topology as well.The algebraic approach, while falling short of Fermat's last theorem,was an outstanding success in the development of algebra as a whole. Inparticular, it was the main stimulus for the development of Dedekind'stheory of ideals.

Fermat's own contribution to Fermat's last theorem was slight, as faras we know. The "marvellous proof" he claimed in his marginal notewas probably based on a mistake, as he did not repeat the claim later,and left only the proof for n = 4 (0.3.1). This proof does not generaliseto other values of n, and it is the only case of Fermat's last theorem thathas been proved by elementary methods.

30 0.5 Roots of unity

The search for an algebraic proof of Fermat's last theorem really beginswith the Algebra of Euler (1770). As we have seen in 0.4.1, this bookcontains a brilliant, and basically sound, treatment of the equation y3 =x2 + 2. It also contains a treatment of x3 + y3 = z3, the Fermat equationfor n = 3. The reasoning is not quite so sound, but it is extremelythought-provoking, and pregnant with possibilities for generalisation.

Euler rewrites the equation as

y3 = z3 - x3

and then factorises the right hand side into

(z-x)(z- (3x)(z-(3x)

where

-1+v/ 53--1 2_2(3 =

are the imaginary cube roots of 1. At this point Euler makes an inter-esting mistake. He wants to argue that the factors z - x, z - S3x, z - saxare relatively prime factors of a cube, hence cubes themselves, but usesthis argument in the wrong setting - in Z[v/-3] rather than Z[(3]. It sohappens that unique prime factorisation fails in Z[-3], as can be seenfrom the example

4=2x2=(1+')(1-"),whose factors all have minimal norm, 4, for nonunit elements of 7G[.However, unique prime factorisation is valid in Z[(3], for geometric rea-sons like those that apply to Z[i] and Z[/], and Euler's idea can bemade to work.

0.5.2 The cyclotomic integersThe numbers Z[(3] in the proof of Fermat's last theorem for n = 3 areexamples of cyclotomic integers. The general ring of cyclotomic integersis Z[(,,,] where

Sn = cos2a

+ i sin21r- -

n n

is a complex nth root of 1. It was first studied by Gauss (1801), forgeometric reasons. The points (n, (n2, ... , (n = 1 are equally spaced

0.5.2 The cyclotomic integers 31

around the unit circle, hence the name "cyclotomic", from the Greek for"circle-dividing". Since

Cn - 1 = ((n - 1)(Cn -1 + ... + (n + 1) and (n :A1

it follows that Cn satisfies the equation

( n-1+...+Cn+1=0,

and hence Z[Cn] can be described explicitly as

Z[Cn] = {ao + a1Cn + ... + an-2Cn-2 ao, a1 i ... , an-2 E Z}.

When n is a prime, zn-1 + + z + 1 = 0 is called the cyclotomicequation. Gauss (1801), article 341, proved that it is the equation ofminimal degree, with integer coefficients, satisfied by Cn. When n is notprime the cyclotomic equation is of degree lower than n - 1, and is notso easy to describe.

Gauss was initially interested in Z[Cn] because of its bearing on theconstruction of the regular n-gon by straightedge and compass, which ispossible when Cn is expressible in terms of rational operations and squareroots. He made the amazing discovery that this occurs for precisely then that are products of a power of 2 with distinct primes of the form22h + 1. The only known primes of this form are 3, 5, 17, 257, 65537,but the latter three yield regular n-gons not previously constructed.In fact, Gauss's construction of the regular 17-gon was the first suchconstruction since ancient times, and was an important factor in hisdecision to become a mathematician.

Perhaps the most surprising feature of this discovery is that primes22h + 1 had already been considered - for an entirely different reason- by Fermat. Fermat mistakenly believed the numbers 22h + 1 to beprime for all values of h (Fermat (1640a)). This is false. Euler foundthat 641 divides 225 + 1, and since then no more such primes have beenfound. Yet in a way Fermat was not far wrong - what is true is that notwo numbers 22h + 1 have a common prime factor. This was observedby Polya and Szego (1924), and used by them to give a new proof thatthere are infinitely many primes. The numbers 22h + 1 are now calledFermat numbers, and the primes among them are called Fermat primes.

Gauss's discovery may have closed a chapter in the history of geom-etry, but it opened a new one in the history of number theory. Quiteapart from drawing attention to the problem of finding all Fermat primes(intractable so far), the theory of cyclotomic integers created new linksbetween different parts of number theory. In particular, Gauss found


connections between arithmetic modulo p, the quadratic integers 7G[ p1

and the cyclotomic integers Z[(]. As we shall explain in the next sec-tion, the connection between cyclotomic integers and quadratic integerscame to light because a solution of the cyclotomic equation by squareroots is needed to construct the regular p-gon.

Cyclotomic integers also arise naturally in connection with Fermat'slast theorem, because of the factorisation

z" - x" _ (z - x)(z - (,,,x) ... (z - x),

which generalises the factorisation used by Euler in his attempt to proveFermat's last theorem for n = 3. In fact, Lame (1847) used this formulain an attempt to prove Fermat's last theorem for arbitrary n > 2. He as-sumed, as would follow from unique prime factorisation, that relativelyprime factors of an nth power are themselves nth powers. Unfortunately,he failed to check uniqueness of prime factorisation in Z[C ], and thisomission turned out to be fatal. Unique prime factorisation is crucial,and it fails in Z[(281. Unbeknownst to Lame, Kummer (1844) had al-ready discovered this, and had taken steps to deal with the problem.However, this is getting ahead of our story. Let us return to Gauss.

0.5.3 Cyclotomic integers and quadratic integersSince a straightedge creates lines (which have equations of degree 1) andthe compass creates circles (which have equations of degree 2), construc-tions find intersections of lines and circles, hence they solve linear andquadratic equations, and this can be done by rational operations andsquare roots. Thus if the regular p-gon is constructible by straightedgeand compass, there is necessarily a solution of the cyclotomic equation

zp-1+...+z+1=0by rational operations and square roots. The roots are gyp, (P, ... , <P-1,hence the equation says that the sum of all the roots is -1. Gauss solvesthe equation by stepwise evaluation of sums of progressively smallersubsets of the roots, now known as Gauss sums. In the case p - 1 = 22heach subset is half the size of the previous one, and its sum is obtainablefrom the preceding Gauss sum by square roots. Thus in 2h steps onereaches a single root of the cyclotomic equation, expressible by squareroots alone.

For example a root C of the cyclotomic equation for p = 5 satisfies

(4+(3+(2+C+1=0,

0.5.3 Cyclotomic integers and quadratic integers 33

which can rewritten

(S3+(2)2+(S3+(2) - 1 =0,

since ((3 + (2)2 = (4 + 2(5 + ( and (5 = 1. Thus we have a quadraticequation for (3 + (2, which can be solved by square roots, and we canthen find 3 and (2 individually by solving quadratic equations, sincewe know their sum and product.

Of course, if p - 1 is not a power of 2 one cannot repeatedly halvethe number of terms in the Gauss sums until only one term remains.However, in the nontrivial case where p is odd, the first halving is alwayspossible. From now on we shall use (, without subscript, to denote anarbitrary root of the cyclotomic equation. The set of all roots (k ispartitioned into those for which k is a square mod p, and those forwhich k is a nonsquare. As we know from the existence of primitiveroots (0.3.5), these sets are of equal size. In fact, we know that thesquares mod p are the even powers of a primitive root mod p, and thenonsquares are its odd powers. In the case p = 5, 2 is a primitive root,so its odd powers are 2 and 23 = 8 - 3 (mod 5), and the correspondingGauss sum is the one used above: (3 + (2. We could also have used itscomplement, S4 + (, whose exponents are the squares mod 5.

In the Disquisitiones, article 356, Gauss showed that the sums whoseexponents are the squares and nonsquares mod p are the two roots ofthe equation

X +xf =0,2p41with the - sign when p - 1 (mod 4) and the + sign when p = 3 (mod4). Since the roots are -2 ± 4 in the former case and -2 ±i$ in thelatter, it follows that, if a is a primitive root mod p,

(a2i (a2j+1 ±Vf-

iP(azi - Ca29+1

=

when p - 1 (mod 4),

when p - 3 (mod 4).i

The left hand sides of these equations can be written more concisely withthe help of the Legendre symbol or quadratic character symbol definedby

k +1 if k is a square mod pp -1 if k is a nonsquare mod p.


Quadratic characters have the following multiplicative property,

\p/ \p/ -(k

pl/ '

which follows immediately from the fact that the squares mod p areprecisely the even powers of a primitive root.

In terms of quadratic characters, the Gauss sum on the left of theequations is simply S = EP-1 (p) (k, and Gauss's theorem is equivalentto:

S+2 _ f +p if p = 1 (mod 4)-p if p 3 (mod 4).

Even more concisely,

S2

\ p /psince we know from 0.3.5 that -1 is a square mod p if and only if p = 1(mod 4).

The proof that S2 = ( i) p goes as follows. Since

P-1-1 ( lSk-1 \ k) (k,

P-1

S2 = ? \p/()Ck+t

k,l-1P-1 klE ck+l

k,1=1 p

by the multiplicative property of quadratic characters.Now each k 0- 0 (mod p) has a multiplicative inverse mod p (0.3.1),

hence as k runs through the nonzero congruence classes mod p, so doeskl, for fixed 1 # 0 (mod p). We may therefore replace k by kl, and

P-1 2S2 = ((k1+1kl

k,1=1 p

P-1 S1(k+1)7,(k)

- k,t=1 p

since k is a square mod p t* kl2 is a square mod p

0.5.3 Cyclotomic integers and quadratic integers 35

P-1 P-2 P-1p - 1) (iP + E (k) (t(k+l)

1 p J k=1 p1=11 2

1 +

P1=1 ` p l k=11 -since l(k + 1) for fixed k < p - 1 runs through the

nonzero congruence classes

(p_ 1 -2-2 (kl

= \ p / (p - 1) + k=1 p/(-1)

by the cyclotomic equation

=

(tJ)pp

since half the (P) are +1 and half are -1, by 0.3.5

- p since p - 1 -1 (mod p).p1

It follows that S = f (Pl )p, and hence Z[('P] contains either or

v, P. Thus Gauss's computation forges links between squares mod p,Z[(P] and Z[V/I]. In the next section we shall how Gauss exploitedthese links to give a proof of quadratic reciprocity.

Remark. The relationship of cyclotomic integers with quadratic inte-gers is special, and there is no comparable relationship with, say, cubicintegers. For example, V2- does not belong to any Z[(P]. One wayto see this is to extend Z[(] to the field Q((p), by allowing arbitraryquotients, and to consider automorphisms of fields. Automorphisms ofQ((p) extend functions of the form r2((P) = (P, and hence any two ofthem commute. On the other hand, any field containing r2 and (3has noncommuting automorphisms, extending all permutations of theset { 3 2, (3 3 2, ( 3 2}. We shall not elaborate on these hints, since theideas belong to Galois theory, which Dedekind wishes to avoid. However,they do show that Galois theory is just over the horizon.


0.5.4 Quadratic reciprocity

The quadratic reciprocity law is the following symmetric relationshipbetween the quadratic characters of two odd primes p, q.

(p) (q) _ ( -1 if p,q=3 (mod 4)q p 5l +1 otherwise.

This surprising relationship can be extracted from the qth power of theGauss sum S = EP-=1 (k) (k by using the "mod q binomial theorem"

(x + y)' - xq + yq (mod q).

As mentioned in 0.3.1, this holds for q prime because in that case all thebinomial coefficients (i), ... , (q? 1) are divisible by q. This leads to the"mod q multinomial theorem"

(x1+x2+...+xk)qxi+xz+...+xq (mod q)

by induction on k, when x1i x2, ... , Xk are indeterminates. We are ableto treat the powers (, (2, ... , (p-1 as indeterminates because the mini-mality of the cyclotomic polynomial (0.5.2) means that each element ofZ[C] is uniquely expressible as a sum E'-1 ak(k. Here is the qth powercalculation:

P(pk) q

SqP

(k

PE-1 (k q

)(kq

(mod q)k=1 p

= E (k)c-kq since(,)n

_ (P for any odd power n

_

(p)2

p(kq since (p)2 = 1

k=1

P-'_ ()q) (kq by the multiplicative property

(p)S

0.5.4 Quadratic reciprocity 37

since, for fixed q, kq runs through the nonzero congruence classes. Wenow multiply both sides by S and use SZ = (Pl) S, obtaining

Sq+i\p/ \ P

S (mod q).

The left hand side iss

(S2)2 _ =11 2 p 2P J

and hence we can cancel (P1) p from both sides to obtain

lQ=

C p /a

p2 \p) (mod q).

Finally, Euler's criterion (0.3.5) allows us to replace p-21 by the quadraticcharacter (q) , and since (Pl) = ±1 we have

if q - 1 (mod 4),

if q - 3 (mod 4),

from which the statement of quadratic reciprocity follows, using thevalue of GO , the quadratic character of -1, known from 0.3.5.

Remarks. The law of quadratic reciprocity has probably been proved inmore ways than any other theorem in mathematics except the theoremof Pythagoras. Gauss himself gave eight proofs, of which the above is thesixth (Gauss (1818)). There are shorter or more elementary proofs, butthe one using Gauss sums is especially enlightening, because of the wayit relates squares mod p to quadratic integers and cyclotomic integers.It was certainly an inspiration to Dedekind, and his own proof (§27 ofthe memoir) shows how Gauss's ideas can be transformed with the helpof ideals. Dedekind says he is giving the sixth proof of Gauss, but hesucceeds in eliminating all the computations. This is exactly what heintends to accomplish with the theory of ideals. As he says in §12:

It is preferable, as in the modern theory of functions, to seek proofs basedimmediately on fundamental characteristics, rather than on calculation, andindeed to construct the theory in such a way that it is able to predict theresults of calculation.


0.5.5 Other reciprocity lawsThe appearance of roots of unity in the proof of quadratic reciprocityis very interesting and historically important, for the reasons just men-tioned, but it is not strictly necessary. There are many other proofsinvolving just arithmetic mod p and q. One reason it is easy to avoidcomplex numbers is that the real numbers + 1 and -1 can serve as valuesof the quadratic character (k).

q

If, on the other hand, one wants to define a multiplicative "cubiccharacter" (k)3, with three different values according as k is of theform a3n, a3n+1 or a3n+a (where a is a primitive root mod p), then realvalues of the character will not work. In fact, the obvious values are

1 if k=a3nk if k = a3n+1q 3

C3 if k=a3n+2

Similarly, the obvious values of the "biquadratic character" (k)4 are1, i, -1 and -i. Thus there is a more pressing reason to use roots ofunity when one seeks reciprocity laws for 3rd and 4th powers. The latterwas in fact what Gauss was doing when he developed the theory ofunique prime factorisation in Z[i]. Likewise, Eisenstein (1844) found acubic reciprocity law through investigation of Z[(3].

Kummer even declared that cyclotomic integers were more importantfor their application to higher reciprocity laws than their application toFermat's last theorem. In 1847 he said:

The Fermat theorem is indeed more of a curiosity than a main point in thescience [of number theory]. (Kummer (1847b).)

And a few years later, when he had made progress on reciprocity laws:

Through my investigations in the theory of complex numbers [cyclotomic in-tegers] and their applications to the proof of the Fermat theorem, which Ihave communicated to the Academy of Sciences over the past three years, Ihave succeeded in discovering the general reciprocity law for arbitrary powerresidues, which in the present state of number theory stands as the mainproblem and pinnacle of the discipline. (Kummer (1850b).)

However, he said this when he was still flushed with success over reci-procity - and when his work on Fermat's last theorem was running outof steam. In the long run not everyone agreed with him. When Hilbertsurveyed algebraic number theory in his influential Zahlbericht (Hilbert(1897)), he gave pride of place to Kummer's results on Fermat's last

0.6.1 Definition 39

theorem. Even today, when Kummer's approach to Fermat's last theo-rem has been abandoned, its ideas are at the core of algebraic numbertheory.

0.6 Algebraic integers0.6.1 Definition

The examples of quadratic and cyclotomic integers hint at a generalconcept of algebraic integer, but it is not clear how it should be defined.In fact, it is not even clear how quadratic integers should be defined ingeneral. An example which underlines the subtlety of the concept is

7 L [ ] = {m + of : m, n E Z}.

As already mentioned in 0.5.1, unique prime factorisation fails in Z[vr-3],but it is valid in Z[t;3], where b3 = -1

a. Thus we would be happier if

all members of Z[(3] were classed as integers, and not just the membersof Z[. At the same time, of course, we do not want to admit allthe members of

Q[/] = Is + t/ : s, t E Q}.This would certainly be going too far, since all rational numbers wouldbe included.

A definition which draws the line at just the right place is the onegiven by Dedekind in the introduction to his memoir. He states it ina way that makes it a natural specialisation of the concept of algebraicnumber:

A number 0 is called an algebraic number if it satisfies an equation

on +a10n-1 +a20n-2 +... +an-10+an = 0

with finite degree n and rational coefficients a1, a2, ..., an-1, an. It is called analgebraic integer, or simply an integer, when it satisfies an equation of the formabove in which all the coefficients al, a2, ... , an_ 1, an are rational integers.

This definition has some desirable properties, as we shall see in the nextsection, but it also has one undesirable property: if a is an algebraicinteger then so is V/a-, hence every algebraic integer has a nontrivialfactorisation a = V/'a-v,'a-. This is very different from the behaviour ofordinary integers, and the theory of all algebraic integers is thereforenot very useful in the study of Z, for the reasons given in 0.1.

Dedekind escaped this situation by restricting attention to the integersin a field of finite degree, which can be most simply defined as the closure

40 0.6 Algebraic integers

Q(a) of Q U {a} under rational operations, where a is an algebraicnumber. The integers of Q(a) are simply the algebraic integers in Q(a).When a is an algebraic integer they include all members of Z[a] butsometimes more. For example, the set of integers in Q(/) is notZ[\] but Z[(31. On the other hand, the set of integers of Q((,) isindeed Z[(p]. Kummer, working without a general definition of algebraicinteger, actually defined his integers to be the members of Z[C,], so in asense he was lucky to hit on the "correct" integers.

0.6.2 Basic propertiesDedekind first stated his definition of algebraic integer in his SupplementX, §160, to the second edition of Dirichlet's Vorlesungen 2iber Zahlen-theorie (Dedekind (1871)). There he added that:

It follows immediately that a rational number is an algebraic integer if andonly if it is a rational integer.

This result goes back at least as far as Gauss. In article 11 of theDisquisitiones he assumes the reader knows that a rational solution of amonic polynomial equation with integer coefficients is itself an integer.

It follows that an ordinary integer a does not divide an ordinary integerb, as an algebraic integer, unless a divides b as an ordinary integer.This is obviously crucial when results about ordinary integers are beingderived as special cases of results about algebraic integers. Similarlycrucial properties of algebraic integers are closure under +, - and x.These follow from another property of monic polynomial equations, firstpointed out by Eisenstein (1850).

If f (x) = 0 is a monic polynomial equation with ordinary integer co-efficients and roots a1, a2, ... , an, and if g(al,... an) is any polynomialin the roots with ordinary integer coefficients, then g(al,... an) is alsoa root of a monic polynomial equation h(x) = 0 with ordinary integercoefficients.

Eisenstein's proof uses the theorem of Newton (1665) that symmetricpolynomials in the roots are polynomial functions of the coefficients,applying it to the equation

h(x) = H (x - g(aa(1),... , ac(n))) = 0.all permutations o of 1,2,...,n

Obviously h(x) is monic, with symmetric coefficients, hence it actu-ally has integer coefficients, and its roots include g(a1, . . . , an). Taking

0.6.3 Class numbers 41

g(al, a2) = al +a2, al - a2 or ala2 shows that al +a2, al - a2, ala2all satisfy monic equations with ordinary integer coefficients, and hencethey are algebraic integers.

Dedekind remarks in §17 of the memoir that he wishes to avoid thetheory of symmetric functions. In fact in §13 (and previously in §160of his Supplement X to Dirichlet (1871)) he has already derived clo-sure from the fact that a system of homogeneous linear equations withnonzero solution must have nonzero determinant. These proofs, alongwith the concepts of basis and independence he also introduced, wereimportant in the development of linear algebra as a self-contained disci-pline. Setting the theory of algebraic integers within the theory of fieldsof finite degree gave this development an extra push, because these fieldsare vector spaces of finite dimension over the rationals. The reason issimple: if the minimal rational polynomial equation satisfied by a is ofdegree n, then 1, a, a2, ... , an-l is a basis for Q(a) over Q.

Fields of finite degree are important to the concept of algebraic inte-ger because they admit a suitable concept of norm. Again generalisingthe situation for quadratic and cyclotomic integers, Dedekind definesthe norm N(w) of any number w E Q(a). If a, al, a2, ... , an_l arethe roots of the irreducible monic equation satisfied by a, and O(a)is the rational function of a equal to w, then N(w) is the productof O(a), d(al), ¢(a2), ... , O(an_l), the so-called conjugates of w. AsDedekind explains in §16 and §17 of the memoir, the conjugates of w arealso its images under all the isomorphisms of Q(a) onto other numberfields. The latter fact makes it easy to see the multiplicative property ofthe norm: N(wlw2) = N(wi)N(w2). It turns out that N(w) is alwaysa rational number, and N(w) is a rational integer when w is an integerof Q(a). This means that every integer has a divisor of minimal norm,and hence a factorisation into "primes" (though the factorisation is notnecessarily unique).

0.6.3 Class numbersAs already mentioned in 0.3.3, Lagrange (1773) gave the first rigorousresults bearing on unique prime factorisation of algebraic integers. Hisreduction process for quadratic forms gives an easy way to find theclass number of quadratic forms with negative discriminant, because thereduced forms are inequivalent, apart from some simple exceptions. Theclass number of x2 - cy2 can be redefined and reinterpreted as the classnumber of the field Q(/) (and generalised to any field of finite degree,

42 0.6 Algebraic integers

see §29 of the monograph). Lagrange's computations were extended byGauss (1801), who showed that the class number of Q(om) is 1 for c =-1,-2,-3,-7,-11,-19,-43,-67,-163 (Disquisitiones, article 303).He also conjectured that these are the only negative c for which Q(/)has class number 1, a conjecture which remained open until 1966, whenit was proved by Baker and Stark. (See Baker (1966) and Stark (1967).)An equivalent statement is that these are the only quadratic fields withnegative discriminant and unique prime factorisation.

The situation is more difficult for quadratic fields with positive dis-criminant. There is a similar reduction process, also due to Lagrange,which shows there are only finitely many reduced forms, but it is nolonger clear which of them are equivalent. Thus we do not immediatelyknow the class number, only that it is finite. Gauss made some progress,giving an algorithm in article 195 of the Disquisitiones to decide equiv-alence of reduced forms. Dirichlet (1839) made even greater progress,using analysis to discover a formula for the class number of quadraticfields. While analytic methods are beyond our scope, Dirichlet's resultdoes indicate the depth of the problem. No more elementary approachhas yet been found, and his related theorem on primes in arithmeticprogressions (Dirichlet (1837)) has likewise not been proved in a moreelementary manner.

Apparently, Dirichlet also found an analytic formula for the classnumber of the cyclotomic field Q((,). Kummer (1847a) mentions this,while giving a direct proof that the class number of Q(Cp) is finite.He gives the following specific values for which the class number is 1:p = 5, 7,11,13,17,19. Thus unique prime factorisation holds in thesefields, and the first failure is in Q((23), which Kummer shows to haveclass number 3. Dirichlet never published his class number formula forQ((p), presumably deferring to Kummer, and Kummer (1850a) gave hisown version.

The finiteness of the class number for all fields of finite degree wasfirst proved by Dedekind (1871), in his first account of ideal theory. Theproof is similar to the one given in the memoir below.

0.6..E Ideal numbers and idealsThe long story of unique prime factorisation, lost and regained, can nowbe summarised as follows. Euclid discovered unique prime factorisationin Z, to all intents and purposes, when he showed that a prime divides aproduct only if it divides one of the factors. Fermat saw trouble coming

0.6.4 Ideal numbers and ideals 43

in the behaviour of x2 + 5y2, Lagrange found what the trouble was(class number greater than 1), and Gauss found an elaborate way roundit (composition of forms). However, Gauss's way out was a retreat fromthe direction already opened by Lagrange - the use of algebraic integersin the study of Z.

The first to embrace the algebraic integers, for all their faults, wasKummer. He could see that unique prime factorisation was lost, but hehoped to regain it:

It is greatly to be lamented that this virtue of the real numbers [i.e. the rationalintegers], to be decomposable into prime factors, always the same ones for agiven number, does not also belong to the complex numbers [i.e. the integers ofcyclotomic fields]; were this the case, the whole theory, which is still laboringunder such difficulties, could easily be brought to its conclusion. For thisreason, the complex numbers we have been considering seem imperfect, andone may well ask whether one ought not to look for another kind which wouldpreserve the analogy with the real numbers with respect to such a fundamentalproperty. (Translation by Weil (1975) from Kummer (1844).)

The first sentence is the one quoted by Dedekind in Latin in his in-troduction: "Maxime dolendum videtur ..." (Kummer's paper was oneof the last important mathematical works written in Latin.) Dedekindgoes on to say:

But the more hopeless one feels about the prospects of later research onsuch numerical domains, the more one has to admire the steadfast efforts ofKummer, which were finally rewarded by a truly great and fruitful discovery.

This was the discovery of ideal numbers. The rest of the story can beleft to Dedekind himself. He explains Kummer's ideal numbers withadmirable clarity, and his own distillation of the concept of ideal fromthem. He also gives a candid account of the difficulties en route to uniqueprime factorisation of prime ideals. The concept of ideal is very simple,and so are the concepts of divisor and product, but unfortunately it isnot clear that if ideal b divides ideal a then a = be for some ideal c.Dedekind was stymied by this difficulty for a long time, and here heexplains how he overcame it after several years of struggle. The memoiris not only a superb exposition, but also a rare opportunity to see a greatmathematician wrestling with a problem, with blow by blow commentsup to the final victory.

44 0.7 The reception of ideal theory

0.7 The reception of ideal theory0.7.1 How the memoir came to be written

Dedekind's first exposition of ideal theory, Dedekind (1871), includedproofs of unique prime ideal factorisation and finiteness of the classnumber, together with a very impressive application of the theory: acorrespondence between the quadratic forms of discriminant D and theideals of Q(om), under which the product of ideals corresponds to com-position of forms. Replacement of the complicated and mysterious formsby objects that behaved like integers should have been a revelation, butDedekind's contemporaries were slow to appreciate his achievement.

There was great resistance to the idea of treating infinite sets as math-ematical objects. Dedekind had tried this in 1857 when he introducedcongruence classes in place of specific residues, and again in 1872 withhis definition of real numbers as Dedekind sections. But he was fighting2000 years of tradition (plus a formidable modern opponent, LeopoldKronecker). The "horror of infinity" that had haunted mathematicssince Zeno was not to be dispelled overnight. Most mathematicianswere not even willing to consider a theory based on infinite sets, letalone appreciate its power or elegance.

Nevertheless, Dedekind tried again. The memoir on algebraic integersfirst appeared in instalments in the Bulletin des sciences mathematiqueset astronomiques in 1877. The reason Dedekind chose this unusual outletwas a letter he received from Rudolf Lipschitz in 1876:

Bonn, 11 March 1876Dear Colleague,

To make the purpose of these lines clear, it is necessary to mention a fewthings first. You have perhaps seen that Darboux' Bulletin has brought outa series of my works on the theory of homogeneous functions of an arbitrarynumber of differentials. Herr Darboux invited me to write such a series in1872, leaving the language up to me. The second editor of the Bulletin, HerrHoiiel, who also translated Dirichlet's German works for Liouville's Journal,made a careful French translation, and since Herr Hoiiel has always been veryattentive and obliging, I have remained in touch with him since. As I recentlyhad cause to write to him, I brought up an idea I have long cherished, namelyconcerning your own investigations in the second edition of Dirichlet's lectureson number theory, from §159 onward. In my opinion they are of rare value,and in Germany itself they have not received their full due. Therefore, itwould be highly desirable if you could arrange to give a detailed analysis ofthese investigations in the Bulletin. Herr Hoiiel has now sent a reply, whichI can best convey to you in his own words, and that is the purpose of myletter. He writes: [French text follows, a translation of which is] Our bulletinwould receive with great pleasure a detailed analysis of M. Dedekind's work.

0.7.2 Later development of ideal theory 45

I have communicated the substance of your letter to M. Darboux, who willhimself write to M. Dedekind, and who will in any case be happy to publishhis work. If you have occasion to write to the learned professor, you may tellhim how flattered we will be by his collaboration, and thank him in advancefor anything he is disposed to send us. [End of French text.]

Of course, I cannot know whether you are inclined to undertake such awork. But I hope you will understand the true motive for the steps I havetaken, namely, the keen desire for the mathematical public to learn the truevalue of first rate research.

With cordiality and deep respect,

Yours,

R. Lipschitz

(Translation of Lipschitz (1986), pp. 47-48.)

On April 29 Dedekind wrote a long reply, the first part of which is:

Dear Colleague,

Your letter brought me great and unexpected joy, since for years I had moreor less given up hope of interesting anybody in my general theory of ideals.With the exception of Professor H. Weber in Konigsberg, with whom I haveworked closely as editor of the forthcoming collected works of Riemann, andwho has expressed his intention to acquaint himself with this theory, you arethe first, not merely to show interest in the subject, but also in such a practicalway, that it revives hope that my work may not have been in vain. I thoughtthat the inclusion of this investigation in Dirichlet's Zahlentheorie would bethe best way to attract a wider circle of mathematicians to the field, butlittle by little I have become convinced that the presentation itself is to blamefor the failure of this plan. I can only suppose that the presentation deterredreaders through excessive brevity and terseness, and since autumn I have beenspending my free time, obtained by resigning my three year directorship of thelocal polytechnic, working out a more detailed presentation of the theory ofideals, and I have come so far as to obtain a somewhat improved form of theessential foundation (the content of §163). (Translation of Dedekind's letterto Lipschitz, Lipschitz (1986), pp. 48-49.)

Correspondence about the series of articles continued until 12 August1876, as the details were worked out and Hoiiel was engaged to trans-late Dedekind's manuscript into French. After their publication in theBulletin, the articles were also published as a book (Dedekind (1877)).

0.7.2 Later development of ideal theoryDedekind continued to present new versions of his ideal theory, revisingit substantially for the third (1879) and fourth (1894) editions of Dirich-let's Zahlentheorie, and expanding its interaction with Galois theory. He

46 0.7 The reception of ideal theory

and Weber also made a major application of ideal theory outside numbertheory, with their arithmetic theory of Riemann surfaces (Dedekind andWeber (1882)). This is the great fruit of their collaboration in the publi-cation of Riemann's works - not only a turning point in the developmentof complex function theory, but also the beginning of modern algebraicgeometry. (The "modern theory of functions" cited by Dedekind in §12of the memoir as the inspiration for proofs "based on fundamental char-acteristics, rather than on calculation," is undoubtedly another referenceto Riemann's work.) Dedekind and Weber developed an ideal theory forfunctions, with polynomials playing the role of "rational" integers, andentire algebraic functions playing the role of "algebraic" integers. Onceagain, prime ideals are important, and unique prime factorisation is valid(in fact, somewhat easier to prove).

In the same year, Kronecker (1882) published a long account of hisown theory of fields and integers, which he had been developing for somedecades. His theory included an equivalent approach to unique primefactorisation, though expressed in a very different, and less readable,language. Kronecker was almost diametrically opposite to Dedekind inmathematical philosophy; he opposed nonconstructive methods, infinity,and in particular the use of infinite sets as mathematical objects. Heeven claimed not to believe in irrational numbers. Dedekind favouredthe free use of infinite sets wherever it was necessary or convenient. Hisdefinitions of irrational numbers (as pairs of sets of rationals) and "idealnumbers" (as the sets he called ideals) should have made the convenienceof his position clear, but in fact neither the Kronecker nor the Dedekindapproach attracted an immediate following.

As Artin (1962) said:

Dedekind's presentation is easy to read and elegant for us today, but at thetime it was too modern. Thus the appearance of Hilbert's Zahlbericht inthe Jahresbericht der Deutschen Mathematikervereinigung in 1897 was greetedwith great joy. Hilbert presented all the results known up to that time and,through great simplifications, made the results of Kummer accessible to awider circle of readers. (Translation of Artin (1962), p. 549.)

The Zahlbericht, as its name suggests, was intended to be a report onnumber theory. Hilbert used groups, rings, fields and modules heavily,but with the clear aim of elucidating properties of numbers. In partic-ular, he used ideal theory to simplify Kummer's results on the Fermatproblem.

It was only in the 1920s that Artin and Emmy Noether created mod-ern ideal theory by abstracting the properties of rings that make unique

Acknowledgements 47

factorisation into prime ideals possible, and using these properties asaxioms. The subject then became part of ring theory, and Dedekind'stheories of algebraic numbers and algebraic functions became mere spe-cial cases, at least in theory. In practice, one still finds the best of bothworlds in the theory of algebraic numbers.

AcknowledgementsI am indebted to George Francis for sending me a copy of the Lipschitz-Dedekind correspondence when it was not available in Australia. Thanksalso to Peter Stevenhagen and John McCleary for corrections and com-ments, and to two anonymous reviewers at Cambridge University Pressfor valuable advice.

Clayton, Victoria, Australia John Stillwell

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Cox, D. A. (1989). Primes of the Form x2 + ny2. Wiley.Dedekind, R. (1871). Supplement X. In Dirichlet (1871), 380-497.Dedekind, R. (1872). Stetigkeit and Irrationalzahlen. English translation

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arithmetische Progression, deren erstes Glied and Differenz ganzeZahlen ohne gemeinschaftliche Factor sind, unendliche viele Primzahlenenthalt. Abh. der Konigl. Preuss. Akad. Wiss., 45-81. Also in hisMathematische Werke, volume 1, 313-342.

Dirichlet, P. G. L. (1839). Recherches sur diverses applications de l'analyseinfinitesimal a la teorie des nombres. J. reine angew. Math., 19,324-369. Also in his Mathematische Werke, volume 1, 411-496.

Dirichlet, P. G. L. (1871). Vorlesungen caber Zahlentheorie. Vieweg, secondedition.

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Edwards, H. M. (1977). Fermat's Last Theorem. Springer-Verlag.Eisenstein, F. G. (1844). Beweis des Reciprocitatssatzes fur die cubischen

Reste in der Theorie der aus dritten Wurzeln der Einheitzusammengesetzten complexen Zahlen. J. reine angew. Math., 27,289-310. Also in his Mathematische Werke, volume 1, 59-80.

Eisenstein, F. G. (1850). Uber einige allgemeine Eigenschaften derGleichung, von welcher die Theorie der ganzen Lemniscate abhangt,

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Euler, L. (1744). Theoremata circa divisores numerorum in hac formapaa ± qbb contentorum. Comm. acad. sci. Petrop., 14, 151-181. Also inhis Opera Omnia ser. I, volume 2, 194-222.

Euler, L. (1749). Letter to Goldbach, 12 April 1749. In Euler (1843),493-495.

Euler, L. (1756). Solutio generalis quorundam problematum diophanteorumquae vulgo nonnisi solutiones speciales admittere videntur. Novi comm.acad. sci. Petrop., 6, 155-184. Also in his Opera Omnia ser. I, volume 2,428-445.

Euler, L. (1761). Theoremata arithmetica nova methodo demonstrata. Novicomm. acad. sci. Petrop., 8, 74-104. In his Opera Omnia, ser. I, volume2,531-555.

Euler, L. (1770). Vollstandige Einleitung zur Algebra. English translationElements of Algebra, Springer-Verlag, 1984.

Euler, L. (1843). Correspondance Mathematique et Physique. St. Petersburg.Reprinted by Johnson Reprint Corporation, 1968.

Fermat, P. (1640a). Letter to Frenicle, 18 October 1640. In Fermat (1894),209.

Fermat, P. (1640b). Letter to Frenicle, August(?) 1640. In Fermat (1894),205-206.

Fermat, P. (1640c). Letter to Mersenne, 25 December 1640. In Fermat(1894), 212.

Fermat, P. (1654). Letter to Pascal, 25 September 1654. In Fermat (1894),310-314.

Fermat, P. (1657). Letter to Digby, 15 August 1657. In Fermat (1894), 345.Fermat, P. (1670). Observations sur Diophant. In Fermat (1896), 241-276.Fermat, P. (1894). tBuvres, volume 2. Gauthier-Villars.Fermat, P. (1896). Euvres, volume 3. Gauthier-Villars.Gauss, C. F. (1801). Disquisitiones Arithmeticae. English translation, Yale

University Press, 1966.Gauss, C. F. (1818). Theorematis fundamentalis in doctrina de residuis

quadraticis demonstrationes et ampliationes novae. Comm. Soc. Reg.Sci. Gott. Rec., 4. Also in his Werke, volume 2, 49-64.

Gauss, C. F. (1832). Theoria residuorum biquadraticorum. Comm. Soc. Reg.Sci. Gott. Rec., 7. Also in his Werke, volume 2, 67-148.

Heath, T. L. (1910). Diophantus of Alexandria. Cambridge University Press.Heath, T. L. (1925). The Thirteen Books of Euclid's Elements. Cambridge

University Press, second edition. Reprinted by Dover, 1956.Hilbert, D. (1897). Die Theorie der algebraischen Zahlkorper. Jber. deutsch.

Math. Verein., 4, 175-546. Also in his Gesammelte Abhandlungen,volume 1, 63-363.

Kronecker, L. (1870). Auseinandersetzung einige Eigenschaften derKlassenzahl idealer complexer Zahlen. Monatsber. Konigl. Akad. Wiss.Berlin, 881-889. Also in his Werke I, 271-282.

Kronecker, L. (1882). Grundziige einer Theorie der algebraischen Grossen. J.refine and angew. Math., 92, 1-122.

Kummer, E. E. (1844). De numeris complexis, qui radicibus unitatis etnumeris realibus constant. Gratulationschrift der Univ. Breslau zur

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Jubelfeier der Univ. Konigsberg. Also in his Collected Papers, volume 1,165-192.

Kummer, E. E. (1846a). Letter to Kronecker, 14 June 1846. In Kummer(1975), 98.

Kummer, E. E. (1846b). Zur Theorie der complexen Zahlen. Monatsber.Akad. Wiss. Berlin, 87-96. Also in his Collected Papers, volume 1,203-210.

Kummer, E. E. (1847a). Beweis der Fermat'schen Satzes der Unmogligkeitvon xA + yA = z" fur eine unendliche Anzahl Primzahlen A. Monatsber.Akad. Wiss. Berlin, 132-141, 305-319. Also in his Collected Papers,volume 1, 274-297.

Kummer, E. E. (1847b). Uber die Zerlegung der aus Wurzeln der Einheitgebildeten complexen Zahlen in ihre Primfactoren. J. reine angew.Math., 327-367. Also in his Collected Papers, volume 1, 211-251.

Kummer, E. E. (1850a). Allgemeine Reciprocitatsgesetze fur beliebig hohePotenzreste. Monatsber. Akad. Wiss. Berlin, 154-165. Also in hisCollected Papers, volume 1, 345-357.

Kummer, E. E. (1850b). Bestimmung der Anzahl nicht aquivalenter Classenfur die aus Aten Wurzeln der Einheit gebildeten complexen Zahlen anddie idealen Factoren derselben. J. reine angew. Math., 93-116. Also inhis Collected Papers, volume 1, 299-322.

Kummer, E. E. (1975). Collected Papers. Springer-Verlag.Lagrange, J. L. (1768). Solution d'un probleme d'arithmetique. Miscellanea

Taurinensia, 4, 19ff. Also in his (Euvres, volume 1, 671-731.Lagrange, J. L. (1770). Nouvelle methode pour resoudre les problemes

indetermines en nombres entiers. Mem. de l'acad. roy. sci. Berlin, 24.Also in his Euvres, volume 2, 655-726.

Lagrange, J. L. (1773). Recherches d'arithmetique. Nouv. mem. de l'acad.sci. Berlin, 265ff. Also in his Euvres, volume 3, 695-795.

Lame, G. (1847). Demonstration general du theoreme de Fermat. ComptesRendus, 24, 310-315.

Legendre, A: M. (1798). Essai sur la Theorie des Nombres. Duprat, Paris.Third edition, entitled Theorie des Nombres (1830), reprinted byBlanchard, 1955.

Lipschitz, R. (1986). Briefwechsel mit Cantor, Dedekind, Helmholtz,Kronecker, Weierstraf3 and anderen. Deutsche MathematikerVereinigung.

Neugebauer, O. and Sachs, A. (1945). Mathematical Cuneiform Texts. YaleUniversity Press.

Newton, I. (1665). Of the nature of equations. In Newton (1967), 519-520.Newton, I. (1967). The Mathematical Papers of Isaac Newton, volume 1.

Cambridge Universtity Press.Polya, G. and Szego, G. (1924). Aufgaben and Lehrsatze aus der Analysis.

Springer-Verlag. English translation Problems and Theorems inAnalysis, Springer-Verlag, 1976.

Stark, H. M. (1967). A complete determination of the complex quadraticfields of class-number one. Michigan Math. J., 14, 1-27.

Weil, A. (1974). Two lectures on number theory, past and present. Enseign.Math., 20, 87-110. Also in his Collected Papers, volume 3, 279-302.

Weil, A. (1975). Introduction to Kummer (1975).Weil, A. (1984). Number Theory: an Approach Through History. Birkhauser.

Part twoTheory of algebraic integers

Introduction

In response to the invitation it has been my honour to receive, I propose,in the present memoir, to develop the fundamental principles of thegeneral theory of algebraic integers I published in the second edition ofDirichlet's Vorlesungen fiber Zahlentheorie. Because of the extraordinaryscope of this field of mathematical research, however, I restrict myselfhere to the pursuit of a single goal, which I shall try to clarify in theremarks that follow.

The theory of divisibility of numbers, which is the basis of arithmetic,was established in its essentials by Euclid. At any rate, the fundamentaltheorem that each integer is uniquely decomposable into a product ofprimes is an immediate consequence of the theorem, proved by Euclid,tthat a product of two numbers is not divisible by a prime unless theprime divides one of the factors.

Two thousand years later, Gauss gave, for the first time, an extensionof the notion of integer. The numbers 0, ±1, ±2.... previously goingunder that name, and which I shall call rational integers from now on,were extended when Gauss introducedt complex integers of the forma + b , where a and b are any rational integers. He showed that thegeneral laws of divisibility of these numbers are identical with those thatregulate the domain of rational integers.

The broadest generalisation of the notion of integer is the following.A number 0 is called an algebraic number if it satisfies an equation

=0,

with finite degree n and rational coefficients a1, a2, ... , an_1i an. It is

t Elements, VII, 32.t Theoria residuorum biquadraticorum, II; 1832.

53

54 Introduction

called an algebraic integer, or simply an integer, when it satisfies an equa-tion of the f o r m above in which all the coefficients a1, a 2 ,-- . , an_1i anare rational integers. It follows immediately from this definition that thesum, difference and product of integers are also integers. Consequently,an integer a will be said to be divisible by an integer /3 if a = /3ry, wherery is likewise an integer. An integer a will be called a unit when everyinteger is divisible by e. By analogy, a prime must be an integer a whichis not a unit and which is divisible only by units a and products of theform ea. However, it is easy to see that, in the domain of all integerswe are considering at present, primes do not exist, since every integerwhich is not a unit is always the product of two, or rather any number,of integral factors which are not units.

Nevertheless, the existence of primes and the analogy with the do-mains of rational or complex integers re-emerges when we restrict our-selves to an infinitely small part of the domain of all integers, in thefollowing manner. If 9 is an algebraic number there is, among the in-finitely many equations with rational coefficients satisfied by 0, exactlyone

on +a19n-1 +...+ an-10 +an = 0,

of minimal degree, and which we call for this reason irreducible. If xo,Si, x2, ..., xn_1 denote arbitrary rational numbers, the numbers of theform

0(9) = xo + x19 + x292 + + xn-19n-1,

the set of which we call 0, will also be algebraic numbers, and they enjoythe fundamental property that their sums, differences, products andquotients also belong to the set 0 . I call such a set St a field of degreet n.The numbers 0(9) belonging to a field 1 are now partitioned, followingthe definition above, into two large sets: the set o of integers and thenonintegral, or fractional, numbers. The problem we set ourselves is toestablish the general laws of division that govern such a system o.

The system o is evidently identical with the system of all rationalintegers when n = 1, or with the complex integers when n = 2 and0 =. Certain phenomena which occur in these two special domainso occur again in every domain o of this nature. Above all, the unlimiteddecomposition which prevails in the domain of all algebraic integers isnever encountered in a domain o of the kind indicated, as one easilysees by consideration of norms. If we define the norm of any number

t Dedekind calls it "finite, of degree n". (Translator's note.)

Introduction

p = 0(9) belonging to the field fI to be the product

N(p) = pµ1µ2 ... pn-1,

55

whose factors are the conjugate numbers

p = 0(0), ttl = Y'(01), µ2 = 0(92), ..., An-1 = G(9n-1)

where 9, 91i 92, ... , 9n-1 denote all the roots of the irreducible nth degreeequation, then N(p) will always be, as we know, a rational number, andnever 0 unless p = 0. At the same time, one always has

N(a,3) = N(a)N(,0)

where a and 0 are any two numbers of the field Q. Now if it is an integer,and hence a number in o, then the conjugate numbers µ1,µ2, ... , µn_1will likewise be integers, and so N(p) will be a rational integer.

This norm plays an extremely important role in the theory of numbersin the domain o. In fact, let any two numbers a, /3 in this domainbe called congruent or incongruent relative to the third p, called themodulus, according as their difference ±(a - /3) is or is not divisible byp. Then we can, just as in the theory of rational or complex integers,partition all numbers of the system o into number classes, such that eachclass is the set of all numbers congruent to a given number (representingthe class). And a deeper study shows us that the number of classes(with the exception of the case p = 0) is always finite, equal at mostto the absolute value of N(p). An immediate consequence of this resultis that N(p) = ±1 just in case p is a unit. Now if a number in thesystem o is called decomposable when it is the product of two numbersof the system, neither of which is a unit, it evidently follows from theabove that each decomposable number can always be represented as theproduct of a finite number of indecomposable factors.

This result again corresponds completely with the law holding in thetheory of rational or complex integers, namely that each composite num-ber is representable as the product of a finite number of prime factors.But at the same time this is the point where the analogy, observed un-til now with the old theory, is in danger of being irrevocably broken.In his researches on the domain of numbers belonging to the theory ofcircle division, hence corresponding to equations of the form 9' = 1,Kummer noted the existence of a phenomenon distinguishing the num-bers of that domain from those considered previously. They differ in amanner so complete and so essential as to leave little hope of preservingthe simple laws that govern the old theory of numbers. In fact, whereas

56 Introduction

each number in the domain of rational or complex integers decomposesuniquely into a product of primes, one discovers that, in the numericaldomains considered by Kummer, a number may be representable in sev-eral entirely different ways as the product of indecomposable numbers.Or what amounts to the same thing, one discovers that the indecompos-able numbers lack the characteristic of genuine primes, inasmuch as aprime cannot divide a product of two or more factors without dividingat least one of the factors.

But the more hopeless one feels about the prospects of later researchon such numerical domains,t the more one has to admire the steadfastefforts of Kummer, which were finally rewarded by a truly great andfruitful discovery. That geometer succeeded in resolving all the apparentirregularities in the laws.t By considering the indecomposable numberswhich lack the characteristics of true primes to be products of idealprime factors whose effect is only apparent when they are combinedtogether, he obtained the surprising result that the laws of divisibility inthe numerical domains studied by him were now in complete agreementwith those that govern the domain of rational integers. Each numberwhich is not a unit behaves consistently in all divisibility situations,whether as divisor or dividend, as a prime or as a product of primefactors, actual or ideal. Two ideal numbers, whether prime or composite,which yield two actual numbers when combined with the same idealnumber are called equivalent, and all the ideal numbers equivalent tothe same ideal number form a class of ideal numbers. The set of allactual numbers, considered as a special case of ideal numbers, forms theprincipal class. To each class there corresponds an infinite system ofequivalent homogeneous forms, in n variables and of degree n, whichare decomposable into n linear factors with algebraic coefficients. Thenumber of these classes is finite, and Kummer succeeded in determiningtheir number by extending the principles used by Dirichlet to determinethe number of classes of binary quadratic forms.

The great success of Kummer's researches in the domain of circle di-vision allows us to suppose that the same laws hold in all numerical

f In the memoir: De numeris complexis qui radicibus unitatis et numeris integrirealibus constant (Vrastislaviae, 1844, §8), Kummer said "Maxime dolendum vide-tur, quod haec numerorum realium virtus, ut in factores primes dissolvi possint quipro eodem numero semper iidem sint, non eadem est numerorum complexorum,quae si esset tota haec doctrina, quae magnis adhuc difficultatibus laborat, facileabsolvi et ad finem perduci posset". (See the Introduction, Section 0.6.4, for anEnglish translation of this passage. Translator's note.)

t Zur Theorie der complexen Zahlen (Crelle's Journal, 35).

Introduction 57

domains o of the most general kind considered above. In my researches,the goal of which has been to arrive at a definitive answer to this ques-tion, I began by building on the theory of higher order congruences,since I had previously noticed that the latter theory allows Kummer'sresearches to be shortened considerably. However, while this methodleads to a point very close to my goal, I have not been able to surmount,by this route, certain exceptions to the laws holding in other cases. I didnot achieve the general theory, without exceptions, that I first publishedin the place mentioned above, until I abandoned the old formal approachand replaced it by another; a fundamentally simpler conception focusseddirectly on the goal. In the latter approach I need no concept more novelthan that of Kummer's ideal numbers, and it is sufficient to consider asystem of actual numbers that I call an ideal. The power of this conceptresides in its extreme simplicity, and my plan being above all to inspireconfidence in this notion, I shall try to explain the train of thought thatled me to it.

Kummer did not define ideal numbers themselves, but only the divis-ibility of these numbers. If a number a has a certain property A, to theeffect that a satisfies one or more congruences, he says that a is divis-ible by an ideal number corresponding to the property A. While thisintroduction of new numbers is entirely legitimate, it is nevertheless tobe feared at first that the language which speaks of ideal numbers beingdetermined by their products, presumably in analogy with the theory ofrational numbers, may lead to hasty conclusions and incomplete proofs.And in fact this danger is not always completely avoided. On the otherhand, a precise definition covering all the ideal numbers that may beintroduced in a particular numerical domain o, and at the same time ageneral definition of their multiplication, seems all the more necessarysince the ideal numbers do not actually exist in the numerical domain o.To satisfy these demands it will be necessary and sufficient to establishonce and for all the common characteristic of the properties A, B, C, .. .that serve to introduce the ideal numbers, and then to indicate howone can derive, from properties A, B corresponding to particular idealnumbers, the property C corresponding to their product.t

f The legitimacy, or rather the necessity, of such demands, which must always beimposed with the introduction or creation of new arithmetic elements, becomesmore evident when compared with the introduction of real irrational numbers,which was the subject of a pamphlet of mine (Stetigkeit and irrationale Zahlen,Brunswick, 1872). Assuming that the arithmetic of rational numbers is soundlybased, the question is how one should introduce the irrational numbers and definethe operations of addition, subtraction, multiplication and division on them. My

58 Introduction

This problem is essentially simplified by the following considerations.Since a characteristic property A serves to define, not an ideal numberitself, but only the divisibility of the numbers in o by the ideal number,one is naturally led to consider the set a of all numbers a of the domaino which are divisible by a particular ideal number. I now call such asystem an ideal for short, so that for each particular ideal number therecorresponds a particular ideal a. Now if, conversely, the property A ofdivisibility of a number a by an ideal number is equivalent to the mem-bership of a in the corresponding ideal a, one can consider, in place ofthe properties A, B, C... defining the ideal numbers, the correspondingideals a, b, c,..., in order to establish their common and exclusive char-acter. Bearing in mind that these ideal numbers are introduced withno other goal than restoring the laws of divisibility in the numericaldomain o to complete conformity with the theory of rational numbers,it is evidently necessary that the numbers actually existing in o, andwhich are always present as factors of composite numbers, be regarded

first demand is that arithmetic remain free from intermixture with extraneouselements, and for this reason I reject the definition of real number as the ratioof two quantities of the same kind. On the contrary, the definition or creation ofirrational number ought to be based on phenomena one can already define clearlyin the domain R of rational numbers. Secondly, one should demand that all realirrational numbers be engendered simultaneously by a common definition, andnot successively as roots of equations, as logarithms, etc. Thirdly, the definitionshould be of a kind which also permits a perfectly clear definition of the calculations(addition, etc.) one needs to make on the new numbers. One achieves all of thisin the following way, which I only sketch here:

1. By a section of the domain R I mean any partition of the rational numbersinto two categories such that each number of the first category is algebraically lessthan every number of the second category.

2. Each particular rational number a engenders a particular section (or twosections, not essentially different) in which each rational number is in the first orsecond category according as it is smaller or larger than a (while a itself can beassigned at will to either category).

3. There are infinitely many sections which cannot be engendered by rationalnumbers in the manner just described. For each section of this kind one createsor introduces into arithmetic a special irrational number, corresponding to thesection (or engendered by it).

4. If a, ,3 are any two real numbers (rational or irrational), then one easilydefines a > 03 or a < 33 in terms of the sections they engender. Moreover, one caneasily define, in terms of these two sections, the four sections corresponding to thesum, difference, product and quotient of the two numbers a, 0. In this way thefour fundamental arithmetic operations are defined without any obscurity for anarbitrary pair of real numbers, and one can really prove propositions such as, forexample, f = f , which had not previously been done, as far as I know, inthe strict sense of the word.

5. When defined in this way, the irrational numbers unite with the rationalnumbers to form a domain 91 without gaps and continuous. Each section of thisdomain 91 is produced by a particular number of the domain itself; it is impossibleto engender new numbers in this domain 91.

Introduction 59

as a special case of ideal numbers. Thus if p is a particular numberof o, the system a of all numbers a = pw in the domain o divisible byp likewise has the essential character of an ideal, and it will be calleda principal ideal. The latter system is evidently not altered when onereplaces p by eµ, where a is any unit in o. Now, the notion of inte-ger established above immediately yields the following two elementarytheorems on divisibility:

1. If two integers a = µw, a' = pw' are divisible by the integer µ, thenso are their sum a+a' = p(w+w') and their difference a-a' = µ(w-w'),since the sum w + w' and difference w - w' of two integers w, w' arethemselves integers.

2. If a = pw is divisible by µ, each number aw' = p(ww') divisible bya will also be divisible by p, since each product ww' of integers w, w' isitself an integer.

If we apply these theorems, true for all integers, to the numbers wof our numerical domain o, with p denoting a particular one of thesenumbers and a the corresponding principal ideal, we obtain the followingtwo fundamental properties of such a numerical system a:

I. The sum and difference of any two numbers in the system a arealways numbers in the same system a.

II. Any product of a number in the system a by a number of the systemo is a number in the system a.

Now, as we pursue the goal of restoring the laws of divisibility inthe domain o to complete conformity with those ruling the domain ofrational integers, by introducing ideal numbers and a corresponding lan-guage, it is apparent that the definitions of these ideal numbers and theirdivisibility should be stated in such a way that the elementary theorems1 and 2 above remain valid not only when the number p is actual, butalso when it is ideal. Consequently, the properties I and II should holdnot only for principal ideals, but for all ideals. We have therefore founda common characteristic of all ideals: to each actual or ideal numberthere corresponds a unique ideal a, enjoying the properties I and II.

A fact of the highest importance, which I was able to prove rigorouslyonly after numerous vain attempts, and after surmounting the greatestdifficulties, is that, conversely, each system enjoying properties I and IIis also an ideal. That is, it is the set a of all numbers a of the domaino divisible by a particular number; either an actual number or an idealnumber indispensable for the completion of the theory. Properties I andII are therefore not just necessary, but also sufficient conditions for anumerical system a to be an ideal. Any other condition imposed on the

60 Introduction

numerical system a, if it is not simply a consequence of properties I andII, makes a complete explanation of all the phenomena of divisibility inthe domain o impossible.

This finding naturally led me to base the theory of numbers in thedomain o on this simple definition, entirely free from any obscurity andfrom the admission of ideal numbers.t

Each system a of integers in a field S2, possessing properties I and II,is called an IDEAL OF THAT FIELD.

Divisibility of a number a by a number p means that a is a numberpw in the principal ideal corresponding to the number p and which canbe conveniently denoted by o(p) or op. At the same time it follows fromproperty II or theorem 2 that all the numbers in the principal ideal oaare also numbers in the principal ideal op. Conversely, it is evident thata is certainly divisible by p when all numbers in the ideal oa, and hencea itself, are in the ideal op. This leads us to establish the followingnotion of divisibility, not just for principal ideals, but for all ideals:

An ideal a is said to be divisible by an ideal b, or a multiple of b, andb is said to be a divisor of a, when all numbers in the ideal a are also inb. An ideal p, different from o, which has no divisors other than o andp, is called a prime ideal.t

Divisibility of ideals, which evidently includes that of numbers, mustat first be distinguished from the following notion of multiplication andthe product of two ideals:

If a runs through all the numbers in an ideal a, and Q runs throughall the numbers in an ideal b, then all the products of the form a,3, andall the sums of these products, form an ideal called the product of theideals a, b, which we denote by ab.§

One sees immediately, it is true, that the product ab is divisible by aand b, but establishing the complete connection between the notions ofdivisibility and multiplication of ideals succeeds only after we have van-quished the deep difficulties characteristic of the nature of the subject.This connection is essentially expressed by the following two theorems:

If the ideal c is divisible by the ideal a, then there is a unique ideal bsuch that ab = c.

t It is of course permissible, though not at all necessary, to let each ideal a correspondto an ideal number which engenders it, if a is not a principal ideal.

t Likewise the ideal number corresponding to the ideal ab is said to be divisible bythe ideal number corresponding to the ideal b, and corresponding to a prime idealone has a prime ideal number.

§ The ideal number corresponding to the ideal ab is called the product of the twoideal numbers corresponding to a and b.

Introduction 61

Each ideal different from o is either a prime ideal, or uniquely ex-pressible as a product of prime ideals.

In the present memoir I confine myself to proving these results in acompletely rigorous and synthetic way. This provides a proper founda-tion for the whole theory of ideals and decomposable forms, which offersto mathematicians an inexhaustible field of research. Of all the later de-velopments, for which I refer to the exposition I have given in Dirichlet'sVorlesungen fiber Zahlentheorie and other memoirs still to appear, I haveincluded here only the partition of ideals into classes, and the proof thatthe number of classes of ideals (or of classes of the corresponding forms)is finite. The first section contains only the propositions necessary forthe present goal, extracted from an auxiliary theory, also important forother researches, which I shall expound fully elsewhere. The second sec-tion, which aims to clarify the general notions by numerical examples,can be omitted entirely. However, I have kept it because it may help inthe understanding of the later sections, where the theory of integers inan arbitrary field of finite degree is developed from the above viewpoint.To do this one needs to borrow just the elements of the general theory offields, a theory whose further development leads easily to the algebraicprinciples invented by Galois, which in their turn serve as a basis fordeeper researches into the theory of ideals.

1

Auxiliary theorems from the theory ofmodules

As I have emphasised in the Introduction, we shall frequently have toconsider systems of numbers closed under addition and subtraction. Thegeneral properties of such systems form a theory so extensive that it canalso be used in other researches; nevertheless, for our purposes just theelements of this theory are sufficient. In order to avoid later interruptionto the course of our exposition, and at the same time to make it easier tounderstand the scope of the concepts on which our theory of algebraicnumbers is based, it seems appropriate to begin with a small numberof very simple theorems, even though their interest lies mainly in theirapplications.

§1. Modules and their divisibility1. A system a of real or complex numbers will be called a module whenall the sums and differences of these numbers also belong to a.

Thus if a is a particular number in the module a, all the numbers

a + a = 2a, 2a + a = 3a, . .

a-a=O, 0-a=-a, -a-a=-2a, ..7

and consequently all numbers of the form xa also belong to a, where xruns through all the rational integers, that is, all the numbers

0,fl,±2,±3,....

Such a system of numbers xa itself forms a module, which we denoteby [a]. Consequently, if a module includes a nonzero number then it

62

§1. Modules and their divisibility 63

includes an infinity of different numbers. It is also evident that thenumber zero, which is in each module, forms a module by itself.

2. A module a will be called divisible by the module b or a multiple ofb, and b a divisor of a, when all the numbers in the module a are alsoin the module b.

The zero module is therefore a common multiple of all modules. Also,if a is a particular number in a module a, then the module [a] will be di-visible by a. Moreover, it is evident that any module is divisible by itself,and that two modules a, b divisible by each other are identical, whichwe shall denote by a = b. Finally, if each of the modules a, b, c, a, .. .is divisible by its immediate successor then it is clear that each will bedivisible by all its successors.

3. Let a, b be any two modules. The system m of all the numbersthat belong to both modules will itself be a module. It will be called theleast common multiple of a, b because each common multiple of a, b isdivisible by m.

Indeed, let p, p' be any two numbers in the system in, and hence inboth a and b. Each of the two numbers p ± p' will belong (by 1) notonly to the module a but also to the module b, and hence also to thesystem in, whence it follows that m is a module. Since all members ofthis module m are in a and also in b, m is a common multiple of a, b.Moreover, if the module m' is any common multiple of a, b, and thuscomposed entirely of numbers belonging to both a and b, then (by virtueof the definition of the system m) these numbers will also be in in, thatis, m' is divisible by in.

4. If a becomes equal in succession to all the numbers in a module a,and,3 to all the numbers in a module b, then the system D of all numbersa +,3 will form a module. This module is called the greatest commondivisor of a, b because every common divisor of a, b is also a divisor ofa.

Indeed, any two numbers 6, S' in the system a can be put in the form6 = a + /3, 6' = a'+ 0' where a, a' belong to the module a and ,3, 0' tothe module b, whence

6±6' = (afa')+()3±,3);

and, since the numbers a ± a' are in a and the numbers ,3 ±,3' are in b,the numbers 6 ± 6' also belong to the system a. That is, a is a module.Since the number zero is in every module, all the numbers a = a + 0

64 Chapter 1. Auxiliary theorems from the theory of modules

of the module a and all the numbers ,Q = 0 +,3 of the module b belongto the module D. Consequently, the latter is a common divisor of a andb. Also, if the module D' is any common divisor of a, b, so that all thenumbers in a and all the numbers in b are in D' then (by virtue of 1) allthe numbers a+,3, that is, all the numbers in the module D, also belongto the module D'. Thus D is divisible by D'.

Having carried out these rigorous proofs, we need not explain furtherhow the notions of least common multiple and greatest common divisorcan be extended to any number (even an infinity) of modules. Neverthe-less, it may be useful to justify the terminology chosen, by the followingremark. If a, b are two particular rational integers, m their least com-mon multiple and d their greatest common divisor, it follows from theelements of number theory that [m] will be the least common multiple,and [d] the greatest common divisor, of the modules [a], [b]. In any casewe shall soon see that the number-theoretic propositions relevant to thiscase can also be deduced from the theory of modules.

§2. Congruences and classes of numbers1. Let a be a module. Numbers w, w' will be called congruent orincongruent modulo a according as their difference ±(w - w') is in aor not. Congruence of the numbers w, w' with respect to the module awill be indicated by the notation

w - w' (mod a).

We immediately deduce the following simple propositions, whose proofswe can omit:

If w w' (mod a) and w' - w" (mod a) then w - w" (mod a).If w w' (mod a) and x is any rational integer then xw - xw' (mod a).If w - w' (mod a) and w" = w"' (mod a) then w ± w" - w' ± w"

(mod a).If w w' (mod a) and D is a divisor of a, then w - w' (mod D).If w w' (mod a) and w - w' (mod b) then w - w' (mod m), where

m is the least common multiple of a, b.

2. The first of the preceding theorems leads to the notion of a classof numbers relative to a module a, by which we mean the set of thosenumbers congruent to a particular number, and hence to each other,modulo a. Such a class modulo a is completely determined by giving a

§2. Congruences and classes of numbers 65

single member, and each member can be regarded as a representative ofthe whole class. The numbers in the module a, for example, form sucha class, represented by the number zero.

Now if b is a second module we can always choose in b a finite orinfinite number of numbers

(or) 01) 32, 33, ... ,

in such a way that each number in b is congruent modulo a to one of thesenumbers, and to only one. Such a system of numbers 3r in the moduleb, which are mutually incongruent modulo a, but which represent allclasses having members in b, I call a complete system of representativesof the module b modulo the module a, and the number of numbers or, orof the classes they represent, will be denoted by (b, a), when it is finite.If, on the contrary, the number of representatives Or is infinite, it willbe convenient to assign the value zero to the symbol (b, a). A deeperexamination of such a system (/3r) of representatives now leads to thefollowing theorem:

3. Let a, b be any two modules, with least common multiple m andgreatest common divisor Z. Any complete system of representatives ofthe module b modulo a will at the same time be a complete system ofrepresentatives of the module b modulo m, and for the module a moduloa; consequently we have

(b, a) _ (b, m) = (a, a).

First of all, it is evident that any numbers /3, 0' in the module b whichare congruent modulo a are congruent modulo m, because f3 - /3' is ina as well as in b, and hence also in m. Now, since each number 3 inthe module b is congruent to one of the representatives ,3, modulo a,and hence also modulo m, and since any two different representativesare incongruent modulo a and hence also modulo m; these numbers Nrin b form a complete system of representatives of the module b modulom. The second part is proved in absolutely the same way: since b isdivisible by a, the numbers Nr3are likewise in a and, by hypothesis, theyare incongruent modulo a. And, since each number 6 in a is of the forma + /3, where a is in a and /3 is in b, we have

6 =a+0=f3 (mod a),

and, since /3 and consequently 6 is congruent to one of the numbers /3rmodulo a, the numbers /3r form a complete system of representatives forthe module a modulo a. Q.E.D.


If b is divisible by a then (b, a) = 1, because all numbers in b are - 0(mod a). Conversely, if (b, a) = 1 then b will be divisible by a, since allnumbers in b are congruent to each other and hence - 0 (mod a). Weevidently have at the same time m = b, a = a.

4. If a is a divisor of a and at the same time a multiple of c, moreoverif ,Qr runs through the representatives of b modulo a and if ys runsthrough the representatives of c modulo b, then the numbers ,Qr + ysform a complete system of representatives of the module c modulo a,and consequently

(c, a) = (c, b) (b, a).

In the first place, all the numbers Nr+ys belong to the module c, since,Qr is in b and hence also in c, and lys is likewise in c.

In the second place, they are all incongruent modulo a. In fact if welet /3', /3" be particular values of fir, and let y', y" be particular valuesof 'ys, then the hypothesis /3' + y' = /3" +'y" (mod a) implies, since a isdivisible by b and /3' - 0 (mod b), that y' - -y" (mod b). However,since y', y" are particular members of the series ys, any two of which areincongruent modulo b, we must have y' = y" and hence the hypothesisabove becomes /3' - /3" (mod a). Now since /3', /3" likewise are particularmembers of the series /3r, any two of which are incongruent modulo a,we must have /3' = /3", which proves the assertion above.

In the third place, it remains to see that each number y in c is con-gruent to one of the numbers /3r + ys modulo a. In fact, since eachnumber y is congruent to one of the numbers 'ys modulo b, we canchoose y = /3 + ys, where /3 is a number in the module b. Also, sinceeach of these numbers /3 is congruent to one of the numbers or moduloa, we can choose /3 = a + or, where a is a number in the module a. Wethen have

y=a+'Ys =a+Qr+ys =fir+ys (mod a).

Q.E.D.

5. Let m be the least common multiple, and a the greatest commondivisor, of two modules a, b, and let p, o be given numbers. The systemof two congruences

w - p (mod a), w - o (mod b)

has a common root if and only if

§ 3. Finitely generated modules 67

p = a (mod a),

and all such numbers w form a class of numbers modulo the module m.

If there is a number w satisfying the two congruences then the numbersw - p, w - a will be in a, b respectively, and since the latter are bothin a, the difference p - or of the numbers will likewise be in Z. Thatis, the above condition p - or (mod i) is necessary. Conversely, if thiscondition is satisfied then (by virtue of the definition of a in §1,4) thereis a number a in a and a number 3 in b whose sum a +,3 = p - ar, hencethe number w = p - a = a +,3 satisfies the two congruences. Thus thecondition is also sufficient. Moreover, if w' is a number satisfying thesame conditions as w, then w' - w will also be in both a and b, and hencealso in m, which means that w' = w (mod m). Conversely, each numberw' in the class represented by w modulo m will satisfy the congruences.Q.E.D.

§3. Finitely generated modules1. Let /31i , C 3 2 , N3, ... , ,Q,,, be particular numbers. All the numbers

13= y1 N1 +Y2132+y3133+"'+yn13n,

where yi, y2, y3, .... yn are arbitrary rational integers, evidently form amodule, which we call a finitely generatedt module [131, /32,133, ... , Qn]

The complex of constants /31 i /32,133, ... , /3n will be called a basist of themodule.

This module [/31i /32, ... , /3n] is evidently the greatest common divi-sor of the n finitely generated modules [/31], [ 32], ... , P,,]. It is easy tosee that each multiple of a finitely generated module is itself a finitelygenerated module, but here I confine myself to proving the followingfundamental theorem, which will later have important applications.

2. If all the numbers in a finitely generated module b = [/31,132,. .. , )3n]

can be transformed into the members of a module a by multiplicationby nonzero rational numbers, then the least common multiple m of a

t Dedekind calls them "finite". (Translator's note.)t Note that Dedekind's basis elements need not be independent. However, he re-

quires them to be independent for fields (§15). (Translator's note.)


and b will be a finitely generated module, and one can choose a systemof 2(n + 1)n rational integers a such that the n numbers

µl = a1/3lµ2 = al /31 +a2/32

An = ain)/31 +a2")Q2 + a3(n)03 + ... + a(n)/3n,

form a basis of m, and at the same time

(b, a) m) = a'1a2a3' ... ann),

By hypothesis there are n nonzero fractionsSi 82 83 Snt1, t2,

t3 tn)whose numerators and denominators are rational integers, such that then products

81 82 83 8n/31, -/7 /33, ...7 on

ti t2 t3 tn

belong to the module a. Since members of a module a are changed intoother members of a when multiplied b y rational integers tl, t2, t3, ... , to(§1,1), the products s1/31, 82/32, 83/33, ... , sn/3n likewise belong to a, andif s denotes the absolute value of the product 818283 sn, the numberss/31, 8/32i 8,03, .... s,3n, and consequently all products s/3, belong to themodule a, where /3 denotes any number in the module b.

Now let v be a particular index from the sequence 1, 2, ... , n. Amongthe numbers in the module [,31,/32, ... , /3,] divisible by b let

denote those belonging to module a and hence also to module b, forexample s/3,,. Among these numbers µ'v there will be at least one number

µv = alv)/31 + a2v)/32 + + a(v)/3v,

for which y takes the smallest positive value av(v). One can then seethat, in all the numbers µv, the coefficient y is divisible by av(v). This isbecause one can always put

yv = xvavv) +' 1/v

where xv and y, are rational integers and the latter satisfies the conditiont

0<yv<a'.f Which is the foundation of the theory of division for rational integers.

§3. Finitely generated modules 69

Then if we put

i (v) , (v) i (v)

yl = yl - xval , y2 = y2 - x"a2 , ..., yv-1 = yv-1 - xvav-1

the number

PV - xvµv = Y I31 + y2,32 + + yv-1,3v-1 + y;Qv

belongs to both the module [,31, #2,...,)3.J and the module m, since p'and µv are in m. But since (by definition of the µ") the coefficient ofQv is less than a,(,") and at the same time positive, it is necessary thatyv = 0, and hence that y" = x"a,") be divisible by a,(,"), as required. Atthe same time

AV xvµv = AV-1

becomes a number in [,31, (32i ... , /3v_1] and also in m, or else becomeszero in the case v = 1.

It follows easily that the n numbers µ", obtained by putting v =n, n - 1, . . . , 2, 1 in succession, enjoy the properties enunciated in thetheorem. Each number µ in the module in, that is, each number Mn inboth a and b = [,31,(32, ... , /n], is of the form

A = µn-1 + xn/Ln

where xn is a rational integer and µn_1 is a number belonging to thetwo modules a and [31,02 , ... , /3n_l], and hence also to the module M.Each number µ'n-1 of this nature is of the form

An'-1 = /Ln-2 + xn-l/Ln-1,

where xn_l is a rational integer and µ'n-2 is a number belonging to thetwo modules a and Qn-1], and so on. Finally, each number14 belonging to the two modules a and [01] is of the form

Al = xuILl

where xl is a rational integer. Thus it is proved that each number µ ofthe module m can be represented in the form

µ = xl/Ll + i2µ2 + ... + xn/Ln,

where x1, x2, ... , xn are rational integers. Conversely, since an arbitrar-ily chosen system of rational integers xl, x2i ... , xn certainly produces anumber p in the module in, because µl, p2, ... , µn themselves belong tom, the n numbers µl, /L2,. - -, µn form a basis of the module m.


To prove the last part of the theorem we have to consider all the

numbersx1131+z2132+...+znfn

of the module b for which the rational integers zi, z2, . . . , z, satisfy then conditions

0<z',<a(").We shall show that these numbers ,0', the number of which is evidentlyequal to a' a2 a,(" ), form a complete system of representatives of themodule b modulo m (or a).

In the first place, all the numbers Q' in the module b are incongruentmodulo m. If

4f1 + ... + zn/33 = z',01 + ... + zn'fn (mod a),

then the numbers z' l', z2 , ... , z;' satisfy the same n conditions as thenumbers z'1, z2',. .., zn. Then if the n differences

zn - zn, zn-1 - Zn-11n I II

Z2/ - z2, z1 - z1

are not all zero, let z - be the first of them with a nonzero value,a value which we can assume to be positive by symmetry, and which isalso < aU") since both numbers z' and z' are < a( v). Then the difference

(zi - zi)al + ... + (z' _ zv)QV

is evidently a number p' in a and [01,02 , ... , .3n] for which the coefficientof 3, is positive and < a,(,", contrary to the definition of the numberµ,,. Thus any two different systems of n numbers z'1, z2..... zn, whichsatisfy the conditions above, also produce two numbers 0' in the moduleb which are incongruent modulo a.

In the second place, it is easy to see that an arbitrary number

0 =x101+x202+ ..+znOn

in the module b is congruent modulo a (or m) to one of the numbers,0' since, if z1, z2, ... , zn are given, it is clear that we can successivelychoose n rational integers

xn,xn-1,...) x2ix1

so that the n numbers

zn = zn + ann)xn,

(n) (n-1)zn-1 = zn-1 + an-lxn + an-1 xn-1 ,

§4. Irreducible systems

....................................z2 = z2 + a2n)xn + a2n-1)xn-1 ... +

a2x2,(n) (n-1) iz1 =z1+a1 xn+a1 x1,

satisfy the n conditions 0 < z, < a,(,'). If we now put

a1 =z1/1+z2/32+...+z'On,

we have

71

)3' = 0 +x1/- 1 +x2/ 2 +... +xnAn,

and hence Q - ,3' (mod m). Q.E.D.

W. Irreducible systems1. A system of n numbers al, a2i ... , an will be called an irreduciblesystem, and its members will be called independent, when the sum

a=x1a1+x2a2+"'+xnan

is nonzero for any system of rational numbers x1i x2, ... , xn which arenot all zero. It then follows that any two different systems of ratio-nal numbers x1i x2, ... , Xn produce unequal sums a. In the contrarycase, that is, when there is a system of rational numbers x1i x2, ... , xn,not all zero, for which the sum a is zero, then the system of numbersa1, a2i ... , an will be called reducible, and the numbers themselves willbe called dependent on each other. If one wants to retain this terminol-ogy in the case n = 1, a single number evidently forms a reducible orirreducible system according as it is zero or not. The definition aboveeasily yields the following theorems, whose number can be increasedenormously, on the determinants of rational numbers.

2. If the n numbers a1, a2,. . ., an are independent, then the n num-bers

al = cial + c2a2 + ... + cnan,a2 = ci al +c2a2 + + cnan,.................................a'n = cin)a1 + c2n)a2 +... + c,(nn)an,


whose n2 coefficients c are rational numbers, form an irreducible or re-ducible system according as the determinant

C = E ±C1 C2 ... cnn)

is nonzero or not.

Since al, a2i ... , an are independent, the sum

x1a' +x2a2' +...+xnan =a',

where xl, x2i ... , xn are arbitrary rational numbers, not all zero, cannotvanish unless we simultaneously have

C11x1 + C1 x2 + ... + C(1n)xn = 0,

C22x2 + C2'x2 + ... + C2n)xn. = 0,

Cnxl+C'n'x2+...+Cnn)xn=0,

which is impossible when C is nonzero. Hence the numbers al, a2, ... , anare independent in that case. But if we have C = 0 there is always asystem of rational numbers xl, X 2, ... , xn satisfying the preceding equa-tions, and not all zero. This is seen immediately when all the n2 coeffi-cients c vanish. If this is not the case then, among the minor determi-nants of C that do not vanish there will be one, say

nf C1iC2 ... Cr(r)

of maximal rank r < n such that the minor determinants of higherdegree vanish. In this case, as we know, the last n - r of the equationsabove will be consequences of the preceding r, and we can put these requations in the form

x1 = P'+lxr+1 + ... +pnxn,

rxr = pr+lxr+l + +pnr)xn,

where the r(n-r) coefficients p are rational numbers. Now if we give then - r quantities xr+l, ... , xn arbitrary rational values then not only canwe ensure that they are not all zero, the quantities x1,. .. , xr will likewisetake rational values. Thus we have a system of n rational numbersX17 X21 ... , xn, not all zero, for which the sum a' is zero. Hence in thiscase the n numbers a', a'2, ... , a'n are dependent. Q.E.D.

§4. Irreducible systems 73

3. If the n independent numbers al, a2i ...,an form one basis of amodule a, and a', a2, ... , a' form another, then we have

a" = c1") al + c a2 + ... + c(")an,

where the n2 coefficients c are rational integers whose determinant is±1, and consequently the numbers ai, a2, ... , an are also independent.

In fact, since the numbers a'" are in the module a = [al, a2, .... an]there are in any case n equations of the preceding form, in which thecoefficients c are rational integers. Conversely, since the n numbers aare in the module a = [ai, a2, ... , an] there are also n equations of theform

a" = e1-)a1 + e2(V)a2 + ... + en")an,

whose coefficients e are likewise rational integers. Substituting in themthe first n equations for the n numbers a',,, and bearing in mind thatthe n numbers a form an irreducible system, it follows that the sum

el")c',,, + e,C,,,,`) = 1 or 0

according as the indices v, v' are equal or not. Then the product of thedeterminants

C(n) . E fe' e2 ... 1,

and, since each factor is a rational integer,

C11 ... fe1'e2 ... en(n) = ±1.

Q.E.D.

Conversely, it is clear that [ai, a2, ... , an] = [al, a2i ... , a,,] whenthere are n equations of the form

a" = cl al + ... + c(")an,

where the coefficients c are rational integers whose determinant = ±1.

4. If the n independent numbers ,(31 i ... , form the basis of a moduleb, and if n numbers al, ... , an, forming the basis of a module a, dependon them via n equations of the form

a" =bl"),31+...+bn")/.3n,

where the coefficients b are rational integers whose determinant B isnonzero, then the number of classes is

(b, a) = ±B.


Now each of the numbers,@ , ... , Qn, and hence each number ,Q, of themodule [,31,,32,. .., i3n] can be changed into a member of the module aby multiplying by a nonzero rational number B. It follows, since a isdivisible by b and hence also equal to the least common multiple of aand b, that a has (by §4,3) a basis of n numbers of the form

a/ = aiv)/31 + a(-))32 +... + a(-))3,,,

whose coefficients a are rational integers chosen so that

(b, a) = a' a2 ... a(n) ±a' a2 ... a(n)

Moreover, since the n numbers a1,.. . , an likewise form a basis of themodule a and since (by 2) each of these two systems of n numbers isirreducible, because we assume this of the system Ql...... Qn, we thenhave (by 3) n equations of the form

av = clv)al + ... + cnv)an,

with rational integer coefficients c whose determinant

fc1Jc2 ... cnn) = fl.

By replacing the numbers a1,. .. , an by their expressions above in termsof the n independent numbers ail, ... , . 3n we see, by comparison with thepreceding expressions for the numbers a'' in terms of the same numbers,3i, ... , on, that

aVv) = C(-)b',,, + c(v)b'; + ....+ cnv)b(;L)

and consequently

±a' ...a (nn) = E fci ... c(n) . fbi ... b(n)

Thus we have (b, a) = ±B. Q.E.D.

This important theorem can easily be extended (and even more simplyby means of the theorem below) to the more general case where thecoefficients b are fractional rational numbers. One then obtains thetheorem

(b, a) = ±B(a, b),

and each of the two numbers of classes, (a, b) and (b, a), can be deter-mined by a simple rule involving the determinant B and all its minordeterminants.

5. If only n among the m numbers al, a2i .... an, forming a basis

W. Irreducible systems 75

of the module a are independent, then a has a basis consisting of nindependent numbers a', a'2, ... , ate.

The hypothesis of this theorem will evidently be satisfied wheneverall the m numbers a1,.. . , an are expressible in terms of n independentnumbers W1.... , wn, as

aµ =rlµ)w1

where the system of coefficientsI I ,rl, r2, ..., rn,If If(r) rl , r2 i ..., rn,

... ... ., .,(m) (m) (m)

r1 , r2 , ..., rn

consists of rational numbers, at least one of whose

m(m-1)...(m-n+1)1 2 n

n x n partial determinants R is nonzero. Otherwise, any n of the m num-bers aA would be dependent. Conversely, it follows from the hypothesisof the theorem, that the m numbers aµ can always be expressed in termsof n independent numbers wV, by choosing the latter to be, for example,n numbers among the m numbers aµ which form an irreducible system.Then, since the n + 1 numbers a.., wl, ... , wn are dependent there is anequation, for each index p, of the form

x not all zero. Moreover, since w1, W2, ... , wnare independent, x0 must be nonzero, and consequently aJ, can be rep-resented, in the manner indicated, in terms of the numbers w,,. Finally,since the m numbers aµ include the n numbers w,,, at least one of thedeterminants R will be nonzero.

I shall therefore assume that the m numbers aµ are represented, inthe manner indicated, in terms of the n independent numbers w,,, andI shall show that, no matter how the numbers w are chosen, there is abasis of the module a = [al, a2,. .., am] consisting of n numbers a' ofthe form

at = clv)w1 +C2")w2 + ... + C(-)wv,

with rational coefficients c. To do this I remark first that we can evi-(,dently choose a positive integer k so that the mn products kr,µ) are


integers. If we now put

Wl = k,31, w2 = k,32, ..., wn = k)3n,

and express the numbers aµ in terms of the numbers On it followsthat the module a = [al, 02, , a,,,,] is divisible by the module b =[)31,)32 , ... , ,3n,], and hence it is the least common multiple of a, b. More-over, since the n numbers f3 become the n numbers w, when multipliedby k and the latter, when multiplied by a nonzero determinant R, be-come numbers of the form

x1a1 + x2a2 + ... + xam,

with rational coefficients x, it is clear that each number ,l3 in the moduleb, when multiplied by a nonzero rational, itself becomes a number in themodule a. It follows from this (by §3,2) that the least common multipleof the two modules a, b has a basis consisting of n numbers of the form

av = alv>/31 + a2v), '2+ ... + avv)Nv,

with rational coefficients a for which the product aia2 ann is nonzero.If we now re-express the numbers 3v in terms of the n numbers w wecan conclude that the assertion above is true, which at the same timeproves the theorem.

6. To the preceding proof I add the following remarks. Since the mnumbers a,, form a basis for the module a just as much as the n numbersa',,, there are m equations of the form

aµ = p1µ>a1 +p2µ) a2 + ... + p, an,

and n equations of the form

a'where the 2mn coefficients p and q are all rational integers. By substi-tuting the first expressions in the second, and bearing in mind that then numbers a' are independent, we deduce that the sum

qp' , + q Pv, + ... + q(m)p(,7)= 1 or 0,

according as the m indices v, v' from the series 1,2,. .. , n are equalor not. Then if P denotes any n x n partial determinant formed fromthe system of coefficients (p), and if Q denotes any determinant formedsimilarly from the system of coefficients (q), then we know that the sum

E PQ,


taken over all combinations of n different upper indices, is equal to 1, andconsequently the determinants P have no common divisor. Conversely,this property of the determinants P is necessary if the n numbers a,,,and also the m numbers

aµ = plµ) al + ... + pn ) an

are to form a basis of the module a.

A system of coefficients such as (p) is evidently just a special case ofthe preceding coefficients (r). Now since the n numbers a can likewisebe represented in the form

aV = e1V)wl + e2V )w2 + ... + e(V)wn,

with n2 rational coefficients e whose determinant

Efele2...enn)is nonzero, we get

r(A) = piµ)ev +p2µ)eU + ... +pnµ)e,

Consequently, the two determinants R, P corresponding to systems ofcoefficients (r), (p) satisfy the relation

R = PE.

The problem of finding all the systems (p) corresponding to a givensystem (r) can be solved in the most comprehensive and elegant mannerby generalising a method applied by Gausst in the special case in whichone utilises identities between the partial determinants. However, thiswould lead us too far away from our present position, and I am content tohave shown the existence of a system such as (p). One sees immediately(from 3) that one can derive from it all other systems (p) by compositionwith all possible systems of n2 rational integers with determinant ±1.

In practice, that is, in any case where the coefficients r are givennumerically and, without loss of generality, as integers, we arrive mostpromptly at the goal by a chain of elementary transformations. Theseare based on the evident proposition that a module [al, a2, ... , am] isnot altered when we replace al, for example, by the number al + xa2iwhere x is any rational integer. The partial determinants R° correspond-ing to all combinations of n numbers from the new basis

a1 = al + xa2, a2 = a2, a3 = a3, ... , a°m = am,

t Disquisitiones Arithmeticae, art. 234, 236, 279.


and to the new system of coefficients (r°), coincide in part with thedeterminants R corresponding to the old basis

0 0 0 0 0a1 = a1 - xa1, a2 =a2 3, a3 = a, ... , am = am

They are of the form R° = R1 + xR2, from which we easily deduce thatthe greatest common divisor E of the determinants R is the same asthat of the determinants R°. Thus the determinants R° cannot simul-taneously vanish. We shall now use these basis transformations of themodule a as follows:

The m coefficients rnµ) of the number wn cannot all be zero, sincethen all the determinants R would be zero. Now if two of these co-efficients, say rn and rn, are nonzero, and if Ir,' I >n jraI, then we canchoose a rational integer x such that I rn + xrn I < I rn . t The elementarytransformation above therefore gives us a new basis in which all the mcoefficients rnµ), except the first, rn, remain the same, and this singlecoefficient is replaced by one of smaller absolute value. By repetition ofthis procedure we necessarily arrive at a basis in which all but one ofthe m coefficients of wn are zero. We denote the member of the basisfor which the latter coefficient a(nn) is nonzero by

an =a(n)wl1 +a 2n)w2 + ... + a(n)wn,

and we keep it fixed in all subsequent transformations of the basis. Thepartial determinants corresponding to the actual basis either vanish, orare of the form Sa(nn), where S is an (n -1) x (n - 1) partial determinantcorresponding to an arbitrary combination of n - 1 of the m - 1 mem-bers of the basis other than an, and which is formed from the (n - 1)2coefficients corresponding to wl, w2,..., wn_1. Since the determinants Scannot all be zero, we now proceed with these m-1 members of the basis,treating wn_1 as we previously did wn in working with the m numbersaµ of the original basis. If we continue these transformations then wefinally obtain a basis of a consisting of n numbers ai, a'2, ... , a'n_1, a''of the form

av = a1v)w1 + a2W2 + ... + an

and m - n = s numbers ai , a2 , ... , a9' which are all zero, and which cantherefore be omitted. The n nonzero coefficients a(V) can be taken tobe positive, since a' can be replaced by -a' without alteration of the

t Here again is the same principle which is fundamental in the theory of rationalintegers.

W. Irreducible systems 79

module, and their product a' a' an(n) is evidently the greatest commondivisor E of all the partial determinants R.

In this way we obtain a second proof of the important theorem (5),and at the same time it is evident that, by composition of the successivetransformations and their inversion, we can find the system of coefficients(p) as well as a system of coefficients (q). In fact, one first obtains mequations of the form

aµ = EP(IL)a/ + h(µ)a"V v o 0V

or, since the s numbers a" are zero,

a (w)aF = P(IA) v,V

and since the determinant of each of the substitutions or transformationsis equal to 1, the m x m determinant

h' ... h'p1 pn 1

pim)p(m)

h(m) ...

hsm)=>2PH=1,

since the quantities H are s x s determinants complementary to thedeterminants P, and formed from the system of coefficients (h). Byinversion, one obtains the adjoint determinant

q1 ... q'n k' ks

q(m) ... gnm) k(-) ... k(mTQK=1,

where K denotes the determinant complementary to Q, and if P, Q arecorresponding determinants then we know that H = Q, K = P. At thesame time we obtain n equations of the form

a = %GA)aA

µ

and s equations of the form

ao= ow)aµ = 0.

The latter equations are a new expression of the original suppositionthat only n of the m numbers a, are independent, and we have beenable to base all this study on such a system of s equations.

One can generally shorten the calculation itself by carrying out several


elementary transformations simultaneously. Suppose for example thatm=4, n=2, whence s = 2, and

a1 = 21w1, a2 = 7w1 + 7w2, a3 = 9w1 - 3w2, a4 = 8w1 + 2W2,

giving

r' = 21, ir" =7, r"'=9,

ir""=8

fir)

,

r' 0, r" = 7 r2 - -3 r"" = 2.2= , 2 , 2 - , 2

We obtain the six partial determinants

(R) R1,2 = 147, R1,3 = -63, R1,4 = 42,1 R3,4 = 42, R2,4 = -42, R2,3 = -84,

using the abbreviation

rl r2µ) - r1µ )r2/i) - R

These determinants satisfy the identity

R1,2R3,4 - R1,3R2,4 + R1,4R2,3 = 0.

Now since the smallest nonzero coefficient of w2 is in a4 we form thenew basis

01 = a1 = 21w1, $2 = a2 - 3a4 = -17w1 + w2,/33 = a3 + 2a4 = 25w1 + W2, /34 = a4 = 8W1 + 2W2,

whence, conversely

ai = /3i, a2=#2+304, a3=,33-2,31, a4=04-

Now since w2 has 1 as its smallest nonzero coefficient, in ,32 for example,we form the third basis

'Yi =)31 =21wi, y2= 32=-17W1+w2,'Y3=-,32+,33=42w1, y4=-2,32+04=42wi,

whence, conversely,

$i = yl, 02 = 72, 33 = y2 +'Y3, ,34 = 2'y2 + 74

At this stage, since 72 is the only number in which w2 has a nonzerocoefficient, and since yi is the number, amongst the other three, inwhich wi has its least coefficient 21, we form the fourth basis

61 = y1 = 21wi, 62 = y2 = -17w1 + w2,63=-2y1+y3=0, 64=-2y1+y4=0,


whence, conversely,

7'1 = 61, 72 = 62, 73 = 261 + 63, 74 = 261 + 64 = 0.

Since 63 = 64 = 0 the transformation is completed, and successive sub-stitutions give

al 61 = 61,

a2 661 +762 +364 = 661 +762,a3 -261 -362 +63 -264 = -261 -362,a4 261 +262 +64 = 261 +262,

and conversely

61 a1 21w1,

62 a2 -3a4 -17w, +w2,63 = -2a1 -a2 +a3 +5a4 0,

64 = -2a1 -2a2 +7a4 0.

Since 61i 62, 63, 64 are the quantities which, in the general theory, we havedenoted by ai, a2, ai, a2, we have

pi = 1, pi = 6, pi/ _ -2, P, - 2,(p) i n - 7 m=-3 an- 2p2 - 0, p2 - , p2 , AT .

Thus we obtain, for the determinants proportional to the R,

(P) P1,2

1 P3,4

and likewise

I qi = 1,q2 = 0,

and

(Q)Q1,2

Q3,4

= 7, P1,3 = -3, P1,4 = 2,= 2, P2,4 = -2, P2,3 = -4,

q1 = 0, q1 f - 0, g111 - 0,q2 - 1, q2 - U, q2 3,

= 1, Q1,3 = 0, Q1,4 = -3,= 0, Q2,4 = 0, Q2,3 = 0

Finally, from the systems of coefficients

(h)

f hi = 0,l h2 = 0,

hi = 0,h2 = 3,

hip = 1,h2I _ -2,

hip' = 0,h2/1 = 1,

dan

(k)ki = _2

fkl = _1, k1I - 1 k111 5,

k2- ,2 k2 =-2, k2' -0: k2"=7

we derive the determinants Hµ,µ, = Q µ and K,.,µ, = Pµ,µ comple-mentary to Pµ,µ, and Qµ,µ' respectively,


H1ni na mi m,2 = h'1 h2 - h'1 h2 > nn " n nnH1,3 = h1 h2 - h1 h2(H){ i n n i m i ' wH3,4 = h1h2 - h1 h2, H2,4 = h1 h2 - h1 h2 ,

K1 2 = kiiiki" - kinikiii1 2 1 2K1,3 = kmiku - kiik"

1 2 1 2>(K) kK34=k'k2 -k'i121 2 2

k'1k

2illK2,4=k'1"ki -1 '

n ni in n> H1,4 = h1 h2 - h1 h2,

' mi nn iH2,3 = h'h1111 - h1 h2,

K14-001-0101 2 1 21k2"-kil2oK2,3 k'1 1 '

and the treatment of the example is complete.To conclude, I remark that applying this to the case n = 1 leads to

the fundamental theorem on the greatest common divisor of an arbitrarynumber of rational integers, the foundation for the whole divisibilitytheory of these numbers.

The researches in this first chapter have been expounded in a specialform suited to our goal, but it is clear that they do not cease to be truewhen the Greek letters denote not only numbers, but any objects ofstudy, any two of which a, 3 produce a determinate third element ry =a + ,a of the same type, under a commutative and uniformly invertibleoperation (composition), taking the place of addition. The module abecomes a group of elements, the composites of which all belong to thesame group. The rational integer coefficients indicate how many timesan element contributes to the generation of another.

2Germ of the theory of ideals

In this chapter I propose, as indicated in the Introduction, to explain ina particular case the nature of the phenomenon that led Kummer to thecreation of ideal numbers, and I shall use the same example to explainthe concept of ideal introduced by myself, and that of the multiplicationof ideals.

§ 5. The rational integersThe theory of numbers is at first concerned exclusively with the systemof rational integers 0, ±1, ±2,±3,..., and it will be worthwhile to re-call in a few words the important laws that govern this domain. Aboveall, it should be recalled that these numbers are closed under addition,subtraction and multiplication, that is, the sum, difference and productof any two members in this domain also belong to the domain. Thetheory of divisibility considers the combination of numbers under mul-tiplication. The number a is said to be divisible by the number b whena = bc, where c is also a rational integer. The number 0 is divisible byany number; the two units ±1 divide all numbers, and they are the onlynumbers that enjoy this property. If a is divisible by b, then ±a willalso be divisible by ±b, and consequently we can restrict ourselves tothe consideration of positive numbers. Each positive number, differentfrom unity, is either a prime number, that is, a number divisible onlyby itself and unity, or else a composite number. In the latter case wecan always express it as a product of prime numbers and - which is themost important thing - in only one way. That is, the system of primenumbers occurring as factors in this product is completely determinedby giving the number of times a designated prime number occurs as a

83

84 Chapter 2. Germ of the theory of ideals

factor. This property depends essentially on the theorem that a primedivides a product of two factors only when it divides one of the factors.

The simplest way to prove these fundamental propositions of numbertheory is based on the algorithm taught by Euclid, which serves to findthe greatest common divisor of two numbers.t This procedure, as weknow, is based on repeated application of the theorem that, for a positivenumber m, any number z can be expressed in the form qm + r, whereq and r are also integers and r is less than m. It is for this reason thatthe procedure always halts after a finite number of divisions.

The notion of congruence of numbers was introduced by Gauss.t Twonumbers z, z' are called congruent modulo the modulus m, written

z - z' (mod m),

when the difference z - z' is divisible by m. In the contrary case z and z'are called incongruent modulo m. If we arrange the numbers in classes,with two numbers in the same class§ only if they are congruent modulom, then we easily conclude from the theorem recalled above that thenumber of classes is finite and equal to the absolute value of the modulusm. This also follows from the studies of the preceding chapter, since thedefinition of congruence in Chapter 1 contains that of Gauss as a specialcase. The system o of all rational integers is identical with the finitelygenerated module [1], and likewise the system m of all numbers divisibleby m is identical with [m]. The congruence of two numbers modulo mcoincides with congruence modulo the system in. Thus (by §3,2 or §4,4)the number of classes is (o, m) = ±m.

§6. The complex integers of GaussThe first and greatest step in the generalisation of these notions wasmade by Gauss, in his second memoir on biquadratic residues, when hetransported them to the domain of complex integers x + yi, where x andy are any rational integers and i is, that is, a root of the irreduciblequadratic equation i2 + 1 = 0. The numbers in this domain are closedunder addition, subtraction and multiplication, and consequently wecan define divisibility for these numbers in the same way as for rational

t See, for example, the Vorlesungen fiber Zahlentheorie of Dirichlet.t Disquisitiones Arithmeticae, art. 1.§ The word class seems to have been employed by Gauss first a propos of complex

numbers. (Theoria residuorum biquadraticor-am, II, art. 42.)

§6. The complex integers of Gauss 85

numbers. One can establish very simply, as Dirichlet showed in a veryelegant manner,¶ that the general propositions on the composition ofnumbers from primes continue to hold in this new domain, as a result ofthe following remark. If we define the norm N(w) of a number w = u+vi,where u and v are any rational numbers, to be the product u2 + v2 ofthe two conjugate numbers u + vi and u - vi, then the norm of a productwill be equal to the product of the norms of the factors, and it is alsoclear that for any given w we can choose a complex integer q such thatN(w - q) < 1/2. If now we let z and m be any complex integers, with mnonzero, it follows by taking w = z/m that we can put z = qm+r whereq and r are complex integers such that N(r) < N(m). We can thenfind a greatest common divisor of any two complex integers by a finitenumber of divisions, exactly as for rational numbers, and the proofs ofthe general laws of divisibility for rational integers can be applied wordfor word in the domain of complex integers. There are four units, ±1,±i, that is, four numbers which divide all numbers, and whose normis consequently 1. Every other nonzero number is either a compositenumber, so called when it is the product of two factors, neither of whichis a unit, or else it is a prime, and such a number cannot divide a productunless it divides at least one of the factors. Every composite number canbe expressed uniquely as a product of prime numbers, provided of coursethe four associated primes ±q, ±qi are regarded as representatives of thesame prime number q. The set of all prime numbers q in the domain ofcomplex integers consists of:

1. All the rational prime numbers (taken positively) of the form 4n+3;2. The number 1 + i, dividing the rational prime 2 = (1 + i)(1 - i) _

-i(1 + i)2;3. The factors a + bi and a - bi of each rational prime p of the form

4n + 1 with norm a2 + b2 = p.The existence of the primes a ± bi just mentioned, which follows

immediately from the celebrated theorem of Fermat on the equationp = a2 + b2, and which likewise implies that theorem, can now be de-rived without the help of the theorem, with marvellous ease. It is asplendid example of the extraordinary power of the principles we havereached through generalisation of the notion of integer.

Congruence of complex integers modulo a given number m of the samekind can also be defined in absolutely the same way as in the theory ofrational numbers. Numbers z, z' are called congruent modulo m, written

I Recherches sur les formes quadratiques a coefficients et a indeterminees complexes(Crelle's Journal, 24).


z = z' (mod m), when z - z' is divisible by m. If we arrange the numbersinto classes, with numbers being in the same class or not according asthey are congruent or incongruent modulo m, then the total number ofdifferent classes will be finite and equal to N(m). This follows very easilyfrom the researches of the first chapter, since the system o of all complexintegers x + yi forms a finitely generated module [1, i] and likewise thesystem m of all the numbers m(x + yi) divisible by m forms the module[m, mi], whose basis is related to that of o by two equations of the form

Consequently we have (§4,4)

(o, m) =a b

-b a= N(m).

§7. The domain o of numbers x + y/There are still other numerical domains which can be treated in abso-lutely the same manner. For example, let 0 be any root of any of thefive equations

82+9+1=0, 02+0+2=0,

92+2=0, 92-2=0, 82-3=0,and let x, y be any rational integers. Then the numbers x + y9 form acorresponding numerical domain. In each of these domains it is easy tosee that one can find the greatest common divisor of two numbers by afinite number of divisions, so that one immediately has general laws ofdivisibility agreeing with those for rational numbers, even though therehappen to be an infinite number of units in the last two examples.

On the other hand, this method is not applicable to the domain o ofintegers

w=x+y9

where 0 is a root of the equation

92+5=0,

and x, y again take all rational integer values. Here we encounter thephenomenon which suggested to Kummer the creation of ideal numbers,and which we shall now describe in detail by means of examples.

§ 7. The domain o of numbers x + y 87

The numbers w of the domain o we shall now be concerned with areclosed under addition, subtraction and multiplication, and we thereforedefine the notions of divisibility and congruence of numbers exactly asbefore. Also, if we define the norm N(w) of a number w = x + yO to bethe product x2+5y2 of the two conjugate numbers x±yO, then the normof a product will be equal to the product of the norms of the factors.And if p is a particular nonzero number we conclude, just as before,that N(µ) expresses how many mutually incongruent numbers there aremodulo p. If p is a unit, and hence divides all numbers, then we musthave N(p) = 1 and therefore p = ±1.

A number (different from zero and ±1) is called decomposable when itis the product of two factors, neither of which is a unit. In the contrarycase the number is called indecomposable. Then it follows from thetheorem on the norm that each decomposable number can be expressedas the product of a finite number of indecomposable factors. However,in infinitely many cases an entirely new phenomenon presents itself here,namely, the same number is susceptible to several, essentially different,representations of this kind. The simplest examples are the following.It is easy to convince oneself that each of the following fifteen numbersis indecomposable.

a=2, b=3, c=7;b1=-2+0, b2=-2-0; c1=2+30, c2=2-30;d1=1+0, d2=1-0; e1=3+0, e2=3-0;fi=-1+20, f2=-1-20; 91=4+0, 92=4-0.

In fact, for a rational prime p to be decomposable, and hence of theform ww', it is necessary that N(p) = p2 = N(w)N(w'), and since w,w' are not units we must have p = N(w) = N(w'), that is, p mustbe representable by the binary quadratic form x2 + 5y2. But the threeprime numbers 2, 3, 7 cannot be represented in this way, as one sees fromthe theory of these forms,t or else by a small number of direct trials.They are therefore indecomposable. It is easy to show the same thing,similarly, for the other twelve numbers, whose norms are products of twoof these three primes. However, despite the indecomposability of thesefifteen numbers, there are numerous relations between their products,which can all be deduced from the following:

(1) ab = d1d2, b2 = b1b2, abi = di,

f See Dirichlet's Vorlesungen fiber Zahlentheorie, §71.


(2) ac = ele2, c2 = clc2, aci = ei,(3) be = fif2 = glg2, a.fi = dlel, agl = d1e2

In each of these ten relations, the same number is represented in two orthree different ways as a product of indecomposable numbers. Thus onesees that an indecomposable number may very well divide a productwithout dividing any of its factors. Such an indecomposable numbertherefore does not possess the property which, in the theory of rationalnumbers, is characteristic of a prime number.

If we imagine for a moment that the fifteen preceding numbers arerational integers then, by the general laws of divisibility, we easily deducefrom the relations (1) that there are decompositions of the formt

a = µa2, dl = µarn, d2 = µa/32,b = µ/31,a2, bl = /t,al, b2 102,

and from the relations (2) that there are decompositions of the form

a =µ'a2, e1 = µ'a'-yl, e2 = µ'a'y2,c = 127172, c1 = µ''yi, c2 = /U' Y2,

where all the Greek letters denote rational integers. And it follows im-mediately, by virtue of the equation µa2 = µ'a'2, that the four numbersfi, f2, gi, g2 appearing in the relations (3) will likewise be integers.These decompositions are simplified if we make the additional assump-tion that a is prime to b and c, since this implies p = p' = 1, a = a'and hence the fifteen numbers can be expressed as follows, in terms offive numbers a, /31, /32, yl, y2:

t Since these decompositions do not seem obvious to me, I include the followingproof of the consequences of (1) as an example. Note first that abi = d2 andb1 b2 = b2 are both squares. Suppose that

a = pat, bh = µ1t1, b2 = p202,

where µ, Al, 92 are squarefree. Then abl = µµ1a2(31 is not a square unlessIA = µl. Similarly, b1b2 is not a square unless Al = µ2. Thus in fact µ = µl = p2and hence

a=µa2, bh =u,Qi,

Forming products of these, we get2 = 2 2 2dl - abl = p a tl

d2 = ab2 =µ2a212b2 = bhb2 = p2t1t2

b2 = µ(32

dl =path,d2 = µa(32,

b = ptlt2,which completes the proof of the decompositions claimed by Dedekind. (Transla-tor's note.)

(4)

§8. Role of the number 2 in the domain o 89

a=a2, b=,31/32, c=yi'Y2,bl = Ql , b2 = /2 i cl = 'Yl , c2 = Iidl = a/31, d2 = a,32; el = aryl, e2 = ar'2iA =0171, f2 = /3272; 91 = /3172, 92 = 327'1

Now even though our fifteen numbers are in reality indecomposable, theremarkable thing is that they behave, in all questions of divisibility inthe domain o, exactly as if they were composed, in the manner indicatedabove, of five different prime numbers a, Nl, /32, 'Y1, 'Y2 In a moment Ishall explain in detail what these relations between numbers mean.

§8. Role of the number 2 in the domain oLet me begin by remarking that, in the theory of rational integers, onecan recognise the essential constitution of a number without effectingits decomposition into prime factors, observing only how it behaves asa divisor. If we know, for example, that a positive number a does notdivide a product of two squares unless at least one of the squares isdivisible by a, then we can conclude with certainty that a is either 1,a prime or the square of a prime. It is likewise certain that a numbera must contain at least one square factor, other than unity, when wecan prove the existence of a number not divisible by a, whose square isdivisible by a. Thus if we can ascertain that both these two propertieshold for a, then we can conclude with certainty that a is the square ofa prime number.

We shall now examine the behaviour, in this sense, of the number 2 inour domain o of numbers w = x + y9. Since any two conjugate numbersare congruent modulo 2 we have

w2 - N(w) (mod 2),

and hence also w2w'2 = N(w)N(w') (mod 2). Now, if the number 2 is todivide the product w2wj2, and hence also the product of the two rationalnumbers N(w), N(w'), it is necessary for at least one of these norms, andhence also for at least one of the two squares w2, w'2, to be divisible by2. Moreover, if we take x, y to be any two odd numbers, then we obtaina number w = x + yO not divisible by 2, and whose square is divisibleby 2. Having regard to the preceding remarks on rational numbers, wethen say that the number 2 behaves in our domain o as though it werethe square of a prime number a.


Although such a prime number a does not actually exist in the domaino, it is by no means necessary to introduce it, since in fact Kummermanaged in similar circumstances with great success by taking such anumber a to be an ideal number, and we are allowing ourselves to beguided by analogy with the theory of rational numbers to define thepresence of the number a in terms of existing numbers w of the domaino. Now, when a rational number a is known to be the square of a primea we can easily judge, without bringing in a, whether a is a factor of anarbitrary rational integer z, and how many times. It is clear that z isdivisible by an if and only if z2 is divisible by a''2. Thus we extend thecriterion to the case we are interested in by saying that a number w ofthe domain o is divisible by the nth power an of the ideal prime numbera when w2 is divisible by 2''2. Experience will show that this definition isvery luckilyt chosen, because it leads to a mode of expression in perfectharmony with the laws of the theory of rational numbers.

It follows first, for n = 1, that a number w = x + y9 is divisible by aif and only if N(w) is an even number, and consequently

(a) x = y (mod 2).

The number w is not divisible by a when N(w) is an odd number, andconsequently x - l+y (mod 2). From this we get the theorem expressingthe character of the ideal number a as a prime number:

"The product of numbers not divisible by a is also not divisible bya."

As far as higher powers of a are concerned, we first conclude fromthe definition that a number w divisible by an is also divisible by alllower powers of a, because a number w2 divisible by 2n is also divisibleby all lower powers of 2. We now have to find, when w is nonzero, thehighest power a"° of a that divides w, that is, the highest power of 2that divides w2. Let s be the exponent of the highest power of 2 thatdivides w itself. We have

w = 28w1 = 28(x1 + y19),

and at least one of the two rational integers x1, yl will be odd. If bothare odd, w1 will be divisible by a and we shall have

W1 = xl - 5y1 + 2x1y19 = 2w2,

f Luckily, since, for example, trying analogously to determine the role of the number2 in the domain of numbers x + y/ leads to complete failure. Later we shallclearly see the reason for this phenomenon.

§9. Role of the numbers 3 and 7 in the domain o 91

where w2 = x2 + Y20 is not divisible by a, because x2 is even and y2 isodd. But if one of the two numbers x1, yl is even, so that the other isodd, then wl and consequently wi will not be divisible by a. Thus inthe first case m = 2s + 1, in the second case m = 2s, but in both casesw2 = 2''nw' where w' is a number not divisible by a. We see at the sametime that m is also the exponent of the highest power of 2 that dividesthe norm N(w). We therefore have the theorem:

"The exponent of the highest power of a that divides a product isequal to the sum of the exponents of the highest powers of a that dividethe factors".

It is likewise evident that each number w divisible by a2n is alsodivisible by 2n since, if the exponent denoted above by s is < n, thenthe numbers 2s, 2s + 1 and hence also m will be < 2n, contrary tohypothesis. It follows immediately from the definition that, conversely,each number divisible by 2n is also divisible by a2n.

Since the number 1 + 0 is divisible by a, but not by a2, we easilysee, with the help of the preceding theorem, that the congruence w2 = 0(mod 2n) that serves to define divisibility of the number w by an can bereplaced by the congruence

(an ) w(1 + 9)n = 0 (mod 2n),

which has the advantage of containing the number w only to the firstpower.

§9. Role of the numbers 3 and 7 in the domain oWhen all the quantities appearing in the equations (4) of §7 are rationalintegers, and if at the same time a is prime to b and c, then it is evidentthat a rational integer z will be divisible by th, /32i 'yi, y2 according asit satisfies the corresponding congruences

zd2 = 0, zdl = 0 (mod b),

zee = 0, zel = 0 (mod c).

These congruences have the peculiarity that they do not involve thenumbers ,31, ,Q2, yl, 'Y2 themselves, and it is for precisely this reasonthat they are appropriate for introducing the four ideal numbers,31, 02,yl, y2 in the context of numbers in the domain o. Wed say that a number


w = x + y9 is divisible by one of these four numbers if w is a root of thecorresponding congruence

(1- 9)w 0, (1 + 9)w 0 (mod 3),

(3 - 9)w 0, (3 + 9)w 0 (mod 7).

Multiplication converts these congruences to the following:

(/31) x - y (mod 3),(/32) x -y (mod 3),(y1) x - 3y (mod 7),(72) x - -3y (mod 7),

concerning which we add the following remarks.Each of these conditions can be satisfied by one of the numbers w =

1 + 9, 1 - 9, 3 + 9, 3 - 9, and the number in question does not satisfyany of the other three, so that it is legitimate to say that the four idealnumbers are all different. Moreover, since every number w divisible byQ1 and ,132 is also divisible by 3, since x =- y - -y - 0 (mod 3) in thatcase, and since conversely every number divisible by 3 is also divisibleby N1 and N2, we ought to regard 3 as the least common multiple of theideal numbers i31 i 02, by analogy with the theory of rational numbers.But each of these two ideal numbers also has the character of a primenumber, that is, it does not divide a product ww' unless it divides atleast one of the factors w, w'. In fact if we put

w=x+y9, w'=x'+y'9, w"=ww'=x"+y"9,then we have

x"=xx' -5yy, y"=xy' + yx',

and hence

x ± y" = (x ± y) (x ± y') (mod 3),

which immediately justifies our assertion, bearing in mind the congru-ences (01), (/32) above. Because of this, the number 3 ought to be con-sidered, from a certain point of view, as the product of the two differentideal prime numbers /31, /32.

Moreover, since each of the ideal prime numbers /31, /32 is different (inthe sense indicated above) from the ideal prime number a introducedabove, then in view of the fact that 2 behaves like the square of a and1 + 9 is divisible by a and /31i and 1 - 9 by a and /32, we ought toconclude from the equation 2.3 = (1 + 0) (1 - 9) that 1 + 0 behaves like

§10. Laws of divisibility in the domain o 93

the product of a and ,(31 i and 1- 0 like the product of a and /32. In factthis presumption is plainly confirmed: each number w = x + y0 divisibleby 1 + 0 is in fact divisible by a and 31, because

x+y0=(1+0)(x'+y'0)

implies

and consequently

x=x'-5y',

x =- y (mod 2),

y=x'+y',

x =- y (mod 3).

Conversely, each number w = x + y0 divisible by a and '31, that is, sat-isfying the two preceding congruences, is also divisible by 1 + 0, becausewe have y = x + 6y' and consequently

x+y0= (1+0)(x+5y'+y'0).

We can now also introduce the powers of the ideal prime numbers ,131,,132, as we have done above for powers of the ideal number a. By analogywith the theory of rational numbers, we define divisibility of an arbitrarynumber w by flln or 02n by the respective congruences

//pi)

Q2 )

w(1-0)n -0 (mod 3n),w(1 + 0)n - 0 (mod 3'' ),

and this yields a series of theorems which agree perfectly with those ofthe theory of rational numbers. We treat the ideal prime numbers y1,rye in the same way.

§10. Laws of divisibility in the domain oBy similar study of the whole domain o of numbers w = x + y0 we findthe following results:

1. All the positive rational primes - 11, 13,17,19 (mod 20) behavelike actual prime numbers.

2. The number 0 with square -5 has the character of a prime number.The number 2 behaves like the square of an ideal prime number a.

3. Each positive rational prime - 1, 9 (mod 20) can be decomposedinto two different factors, which really exist and have the character ofprimes.


4. Each positive rational prime - 3,7 (mod 20) behaves like theproduct of two different ideal prime numbers.

5. Each actual number w different from zero and ±1 is either one ofthe numbers mentioned above as having the character of a prime, or elseit behaves in all questions of divisibility as a unique product of actualor ideal prime factors.

However, to arrive at this result and to become completely certainthat the general laws of divisibility governing the domain of rationalnumbers extend to our domain o with the help of the ideal numbers wehave introduced,t it is necessary, as we shall soon see when we attempta rigorous derivation, to make a very deep investigation, even supposingknowledge of the theory of quadratic residues and binary quadratic forms(a theory which, conversely, can be derived with great facility from thegeneral theory of algebraic integers). We can indeed reach the proposedgoal with all rigour; however, as we have remarked in the Introduction,the greatest circumspection is necessary to avoid being led to prematureconclusions. In particular, the notion of product of arbitrary factors,actual or ideal, cannot be exactly defined without going into minutedetail. Because of these difficulties, it has seemed desirable to replacethe ideal number of Kummer, which is never defined in its own right,but only as a divisor of actual numbers w in the domain o, by a noun forsomething which actually exists, and this can be done in several ways.

One can, for example (and if I am not mistaken, this is the way chosenby Kronecker in his researches), replace the ideal numbers by actualalgebraic numbers, not from the domain o, but rather adjoined to thisdomain in the sense of Galois. Indeed, if we put

/31 = -2 + 0, /32 = -2 - 0,

and if we choose the square roots so that /3102 = 3 then we have

92 = -5, /31 = -2 + 9, 02 = -2 - 9,

/31/32 = 3, 9/31 = -201 - 3,32, 902 = 301 + 202,

whence it follows that the quadrinomial numbers

x+YO +z1/31+z2/32,

t To some people it seems evident a priori that the establishment of this harmonywith the theory of rational numbers can be imposed, whatever happens, by theintroduction of ideal numbers. However the example, given above, of the irregularrole of the number 2 in the domain of numbers x + y suffices to dispel thisillusion.

§11. Ideals in the domain o 95

where x, y, z1, z2 are rational integers, are closed under addition, sub-traction and multiplication. The domain o' of these numbers containsthe domain o, and all the ideal numbers needed for the latter can bereplaced by actual numbers of the new domain o'. For example, byputting

a=,31+/32, 7'1 =2,31+/32, 7'2 =)31+2,32

all the equations (4) of §7 are satisfied. Likewise, the two ideal primefactors of the number 23 in the domain o are replaced by the two actualnumbers 2,31 - /32 and -,31 + 2,32 of the domain o', and it is the samefor all the ideal numbers of the domain o.

Although this way is capable of leading to our goal, it does not seemto me as simple as desirable, because one is forced to pass from the givendomain o to a more complicated domain o'. It is also easy to see thatthe choice of the new domain o' is highly arbitrary. In the IntroductionI have explained in detail the train of thought that led me to build thistheory on quite a different basis, namely on the notion of ideal, and itwould be superfluous to come back to it here; hence I shall confine myselfto illustrating the notion by an example.

§11. Ideals in the domain oThe condition for a number w = x + y9 to be divisible by the idealprime number a is that x = y (mod 2), by §8. Thus to obtain thesystem a of all numbers w divisible by a we put x = y + 2z, where y andz are arbitrary rational integers. The system a therefore consists of allnumbers of the form 2z+(1+9)y, that is, a is a finitely generated modulewith basis consisting of the two independent numbers 2 and 1 + 0, andconsequently

a= [2,1 + 9].

Similarly letting bi, b2, ci, c2 denote the systems of all numbers w divis-ible by ,31i Q2, -ti, rye respectively, we conclude from the correspondingcongruences in §9 that

b1=[3,1+0], b2=[3,1-01,ci = [7,3+8], c2 = [7,3-9].

If we now let m denote any one of these five systems, then m enjoysthe following properties.


I. The sum and difference of any two numbers in m are also in m.II. Each product of a number in the system m by a number in the

system o is a number in the system m.The first property, characteristic of each module, is evident. To es-

tablish the second property for a system m whose basis consists of thenumbers p, µ' it evidently suffices to show that the two products Op, 9µ'belong to the same system. For the system a this follows from the twoequations

20=-1.2+2(1+0), (1+6)6=-3.2+(1+0),

and it is just the same for the other systems. But these two propertiescan also be established without these verifications, by appealing to thefact that each of the five systems m is the set of all numbers w in thedomain o satisfying a congruence of the form

vw - 0 (mod a),

where a, v are two given numbers in the domain o.We now call any system m of numbers in domain o enjoying properties

I and II an ideal, and we begin by posing the problem of finding thegeneral form of all ideals. Excluding the singular case where m consistsof the single number zero, we choose an arbitrary nonzero number it inin. Then if u' denotes the conjugate number, the norm N(µ) = µµ',and hence the product 9N(i) also belongs to the ideal m by virtue of II.Thus all the numbers in the module o = [1, 0], when multiplied by thenonzero rational number N(µ), become numbers in the module in, whichis at the same time a multiple of o. But it is a consequence of (§3,2) thatm is a finitely generated module, of the form [k, l +m9] where k, 1, m arerational integers, among which k and m can be chosen positive. Since malready has property I, as a module, the question is what follows fromproperty II, which says that the two products k9 and (l + mG)9 belongto the system m. The necessary and sufficient conditions for this, as onesees without difficulty, are that m divide k and 1 and that the rationalintegers a, b appearing in the expression

m = [ma, m(b + B)]

also satisfy the congruence

b2 -5 (mod a).

If we replace b by any number b (mod a) then the ideal m is unchanged.

§11. Ideals in the domain o 97

The five ideals above, a, b1i b2, c1, c2 are evidently of this form, since(b + 9) can also be replaced by -(b + 0).

The set of all numbers conjugate to the numbers in an ideal m isevidently also an ideal

ml = [ma, m(-b + 0)].

Two such ideals in, m1 may be called conjugate ideals.Let p be any number in the domain o. The system [p, p9] of all

numbers divisible by p forms an ideal which we call a principal ideal,tand which we denote by o(p) or op. It is easy to give it the aboveform [ma, m(b + 9)]; m is the greatest rational integer that divides p =m(u + v9) and we have, moreover,

a= N(2), vb-u(mod a).

Thus we find, for example,

o(±1) = o = [1, 9],

and

o(2) = [2,20], o(3) = [3,30], o(7) = [7,70],

o(l±0) = [6,±1+0], o(3±9] = [14,±3+9],o(-2 ± 0) = [9,::F2 + 0], o(2 ± 39) = [49, ±17 + 9],

o(-1 ± 20) = [21, ±10 + 01, o(4 ± 0) = [21,±4 + 0].

Since all ideals are also modules, we say (following §2,1) that two num-bers w, w' are congruent modulo the ideal in, and put w - w' (mod m)when w - w' is a number in in. The norm N(m) of the ideal m =[ma, m(b + 0)] is the number

(0,M) = m2a

of classes into which the domain o is partitioned modulo the module m(§4,4). If m is a principal ideal op then the preceding congruence willbe equivalent to w - w' (mod p) and we shall have

N(m) = N(p).

The norm of any number m(ax+(b+9)y) in the ideal m = [ma, m(b+0)]

f If we extend the definition of ideal to the domain o of rational integers, or to thecomplex integers of Gauss, or to any of the five domains o considered in §7, thenone easily sees that every ideal is a principal ideal. It is also evident that, in thedomain of rational integers, property II is already contained in property I.


is equal to the product of N(m) = m2a with the binary quadratic formaxe+2bxy+cy2 whose determinant, according to the definition of Gauss,isb2-ac=-5.$

§12. Divisibility and multiplication of ideals in oI shall now show how the theory of numbers w = x + y9 in the domainm can be based on the notion of ideal. For the sake of brevity, however,I shall be obliged to leave certain easy calculations to the reader.

Just as in the theory of modules (§1,2), we say that an ideal m" isdivisible by an ideal m when all numbers in m" belong to in. It followsthat a principal ideal op" is divisible by a principal ideal op if and onlyif the number µ" is divisible by the number p. Thus the theory ofdivisibility of numbers is contained in the theory of divisibility of ideals.One sees immediately that the necessary and sufficient conditions forthe ideal m" = [m"a", m"(b" + 8)] to be divisible by the ideal m =[ma, m(b + 9)] are the three congruences

m"a-m"a'-m"(b"-b)-0 (mod ma).The definition of multiplication of ideals is the following: if p runsthrough the numbers in the ideal in, and µ' through the numbers inthe ideal m', then all the products µp' and their sums form an idealm" called the productt of the factors in, m' and denoted by mm'. Weevidently have

om = in, mm' = m'm, (mm')n = m(m'n),

whence it follows that products of any number of ideals satisfy the sametheorems as products of numbers.$ Moreover, it is clear that the productof two principal ideals op and oµ' is the principal ideal o(µµ').

Now given two ideals

m = [ma, m(b + 9)], m' _ [m'a', m'(b' + 9)],

we derive their product

m" = mm' = [mam"(b" + 0)]

t The general theory of forms is nevertheless simplified a little when we also admitthe forms Ax2 + Bxy + Cy2, where B is odd, and if we always understand thedeterminant of the form to be the number B2 - 4AC.

t The same definition also applies for multiplication of two modules.t See Dirichlet's Vorlesungen fiber Zahlentheorie, §2.

§ 12. Divisibility and multiplication of ideals in o 99

with the help of the methods indicated in the first chapter (§4, 5 and 6).It is clear from the definition that the product mm' is a finitely generatedmodule with basis consisting of the four products

mm'aa', mm'a(b' + 9), mm'a' (b + 9),

mm'(b + 9)(b' + 9) = mm'[bb' - 5 + (b + b')9],

of which only two are independent. Thus for the ideals considered above,for example

b1=[3,1+9}, C2=[7,3-0],

we find the product

b1c2 = [21,9-39,7+79,8+29].

This module is derived from the one considered at the end of the firstchapter (§4,6), and by setting w1 = 1, w2 = 9 we conclude

b1c2 = [21,-17+9] _ [21,4+9] = o(4+9).

In the same way we obtain all the following results, completely analogousto the hypothetical equations (4) of §7:

0(2) = a2, o(3) = bib2,

o(-2+0) = bi,o(2+30) = ci,

o(1 + 9) = ab1,

o(3 + 9) = act,

o(-1 + 29) = b1c1,

o(4 + 9) = blc2,

o(7) = c1c2;

o(-2 - 9) = b2;

o(2-30) = c2;

o(1 - 9) = ab2;

o(3 - 9) = ac2i

o(-1 - 29) = b2c2;

o(4 - 9) = b2c1.

To effect the multiplication of two ideals; m, m' in general it is necessaryto transform the basis of the four numbers above into one consisting ofonly two numbers m"a", m"(b" + 9). One arrives at this (by virtue of§4) via four equations of the form

mm'aa' = pm"a" + qm"(b" + 0),

mm'a(b' + 9) = p'm"a' + q m"(b" + 9),

mm'a'(b + 9) = p"m"a" + q"m"(b" + 9),

mm'[bb'-5+(b+b')9]=p"m"a"+q"'m"(b"+ 0),

where p, p', ... , q" denote eight rational integers chosen so that the six


determinants formed from them,

P=pq-qp', Q = pq" - qp", R=pq"-qp'",U = pogo, - q'p"', T = pig" - q'p"', S = p'q" - q p"

have no common divisor. From the four preceding equations, each ofwhich decomposes into two, we now conclude without difficulty thatthese six determinants are respectively proportional to the six numbers

a, a', b' + b,

c, c', b' - b,

where c and c' are determined by the equations

bb - ac = b'b' - a'c' = -5.

But, since these six numbers admit no common divisor,t they mustprecisely coincide with the six determinants. It follows, since q = 0and q', q", q"' can have no common divisor, that we can determine theproduct m" = mm' of two given factors m, m' as follows. Let p be thegreatest common (positive) divisor of the three given numbers

a = pq', a' = pq", b + b' = pq"i

We haveas

m" = pmm' a' = 2 = 4q

and b" is determined by the congruences

qb" = q'b', qlb" - q'b, q"ibii = bb' - 5 (mod a").P

At the same time we have b"b" - -5 (mod a"), that is

b"b" - a"c" = -5,

where c" is a rational integer and, to use a terminology employed byGauss,t the binary quadratic form (a", b", c") is composed from the twoforms (a, b, c) and (a', b', c').

The values of m", a" yield mj2a" = and hence the theorem

N(mm') = N(m)N(m').

t This will not always be so in the domain of numbers x + yvl--3.t Disquisitiones Arithmeticae, art. 235, 242.

§12. Divisibility and multiplication of ideals in o 101

It is also necessary to note the special case where m' is the ideal m1conjugate to m. The preceding formulas then yield the immediate result

mm, = oN(m).

The two notions of divisibility and multiplication of ideals are now con-nected in the following manner. The product mm' is divisible by bothm and m' since, by property II of ideals, all the products pp' whosefactors belong respectively to m, m' are in both these ideals; the samethen holds for the product ideal itself. Conversely, if the ideal m" _[m"a", m"(b" + 0)) is divisible by the ideal m = [m, m(b+ 0)] then thereis exactly one ideal m' such that mm' = m". In fact, if we let ml denotethe ideal conjugate to m and form the product

mlm" _ IM///a/, m"'(b' + 0)]

by the preceding rules, then it follows from the three congruences estab-lished at the beginning of this § that m"' is divisible by N(m) = m2 a,and hence that m"' = m2am', where m' is an integer. Combining thiswith the preceding theorem that mm1 = o(m2a), we easily concludethat the ideal m' = [m'a', m'(b'+0)], and it alone, satisfies the conditionmm' = m". At the same time it follows that the equation mm' = mmalways implies m' = m"'.

Now to arrive at the conclusion of this theory, it only remains tointroduce the following notion. An ideal p, different from o and divisibleby no ideals other than o and p, will be called a prime ideal. If 77 is aparticular number, then the system r of all roots p of the congruencerlp - 0 (mod p) forms an ideal, because it has properties I and II. Thisideal r is a divisor of p, because all numbers in p are also roots of thiscongruence. Thus, if p is a prime ideal, then r must be o or p. If thegiven number rl is not in p then the number 1, in o, will not be a rootof the congruence, and hence in this case r will be neither o nor p. Thatis, all the roots p must be in p. Thus we have established the followingtheorem: t

"A product rlp of two numbers rl, p is not in a prime ideal p unless atleast one of the factors is in p."

And this immediately yields the theorem:

t This theorem leads easily to the determination of all prime ideals contained in o,and they correspond exactly to the prime numbers, actual and ideal, enumeratedin §10.


"If neither of the ideals m, m' is divisible by the prime ideal p thentheir product will also not be divisible by p."

Because if m, m' respectively include numbers p, p' not in p, then mm'includes the number pp' not in p.

Combining the theorem just proved with the preceding theorems onthe connection between divisibility and multiplication of ideals, andbearing in mind that o is the only ideal with norm 1, we arrive, byexactly the same reasoningt as in the theory of rational numbers, at thefollowing theorem: "Each ideal different from o is either a prime idealor else uniquely expressible as a product of finitely many prime ideals."It follows immediately from this theorem that an ideal m" is divisible byan ideal m if and only if all the powers of prime ideals that divide m alsodivide m". If m = op and m" = op" are principal ideals then the samecriterion also decides the divisibility of the number p" by the number A.Thus the theory of divisibility of numbers in the domain o is restored tofirm and simple laws.

All this theory can be applied almost word for word to any domaino consisting of all the integers of a quadratic field St, when the notionof integer is defined as in the Introduction.4 However, even thoughthis approach to the theory leaves nothing to be desired in the way ofrigour, it is not at all what I propose to carry out. One notices, infact, that the proofs of the most important propositions depend uponthe representation of an ideal by the expression [ma, m(b + B)] and onthe effective realisation of multiplication, that is, on a calculus whichcoincides with the composition of binary quadratic forms given by Gauss.If we want to treat fields 92 of arbitrary degree in the same way, thenwe shall run into great difficulties, perhaps insurmountable ones. Evenif there were such a theory, based on calculation, it still would not beof the highest degree of perfection, in my opinion. It is preferable, asin the modern theory of functions, to seek proofs based immediately onfundamental characteristics, rather than on calculation, and indeed toconstruct the theory in such a way that it is able to predict the resultsof calculation (for example the composition of decomposable forms of alldegrees). Such is the goal I shall pursue in the chapters of this memoirthat follow.

t See Dirichlet's Vorlesungen fiber Zahlentheorie, §8.t The domain, mentioned above, of numbers x + y/ where x, y are rational

integers is not a domain of this nature. However, it forms only a part of the domaino of all the numbers x + yp, where p is a root of the equation p2 + p + 1 = 0.

3General properties of algebraic integers

In this chapter we first consider the domain of all algebraic integers.Then we introduce the notion of a field SZ of finite degree, and determinethe constitution of the domain o of integers of the field Q.

§13. The domain of all algebraic integersA real or complex number 0 is called algebraic when it satisfies an equa-tion

en+a1en-1+a2en-2+...+an_10+an =0

of finite degree n with rational coefficients al, a2i ... , an_1 i an. If thecoefficients of this equation are rational integers, that is, numbers fromthe sequence 0, ±1, ±2,..., then 0 is called an algebraic integer, or simplyan integer. It is clear that the rational integers are also algebraic integersand, conversely, if a rational number 0 is at the same time an algebraicinteger then, by virtue of a known theorem, it will also be one of therational integers 0, ±1, ±2,.... From the definition of integers we easilyderive the following propositions:

1. The integers are closed under addition, subtraction and multipli-cation, that is, the sum, difference and product of any two integers a, 3are also integers.

Proof. By hypothesis, there are two equations of the form

O(a) = as + piaa-1 + ... + pa-la + pa. = 0,V)(3) _ ,3b + gl,ab-1 + ... + qb-1Q + qb = 0,

in which all the coefficients p, q are rational integers. We now put ab = n

103

104 Chapter 3. General properties of algebraic integers

and let

w1,w2,...,wn

denote the n products as /36' formed from one of the a numbers2 a-11,a,a ,...,a

and one of the b numbers

1, P,Q2...... (36-1

If w now represents one of the three numbers a +,3, a - 0, a/3 then weeasily see that each of the n products ww1i ww2, ... , wwn can be reducedimmediately, with the help of the equations O(a) = 0, 0(/3) = 0, to theform

kiwi +k2w2+...+knwn,

where k1, k2,. .. , kn are rational integers. We then have n equations ofthe form

wwi = kiwi + k2w2 + ... + klwn,

WW2 = ki w1 + k2w2 + ... + knwn,

wwn = k(n')w1 + k(n)w2 + ... + k(n)wn,

all coefficients k of which are rational integers. Now by elimination ofthe n numbers w1, w2, ... , wn, which include the nonzero number 1, wederive the equation

ki - w k2 ... k'n

k1 k- w ... k =0

k(n) k(2n) ... k(n) - w

which is evidently of the form

wn + elwn-1 + ... + en-1w + en = 0,

where the n coefficients e are formed from the numbers k by addition,subtraction and multiplication, and hence are rational integers. Thusw, and consequently each of the three numbers a +,3, a - /3, a$, is aninteger. Q.E.D.

2. Each root w of an equation of the formF(w) =w'n+aw" +Qwm-2+...+E=0,

§14. Divisibility of integers 105

where the coefficient of the highest degree term is unity and the othersa, /3, ... , e are integers, is likewise an integer.

Proof. By hypothesis, the coefficients a, ,(3, . . . , e are roots of equations

O(a)=as+pla' 1+...+Pa =0,

W(Q) _Ob+ql)3b-1+...+qb=0,.................................X(C) = Ee + s1Ee-1 + ... + se = 0,

where all the coefficients p, q, . . . , s are rational integers. Now if we putn = mab e and let w1, w2, ... , wn denote all the n products of theform

wm'aatOb' ... e'

where the exponents are rational integers satisfying

0<m'<m, 0<a'<a, 0<b'<b, ..., 0<e'<e,then it is easily seen that, with the help of the equations F(w) = 0,O(a) = 0, 0(0) = 0, ..., x(e) = 0, all the products WW1, WW2, ... , wwnreduce immediately to the form

k1w1 + k2w2 + ... + knwn,

where k1, k2, ... , kn are rational integers. It then follows, as in the pre-vious proof, that w is an integer. Q.E.D.

It follows from the latter theorem that, for example, if a is any integerand r, s are positive rational integers then B a' is also an integer.

§ 14. Divisibility of integersWe say that an integer a is divisible by an integer ,3 when a = '3-yfor some integer ^y. The same thing is expressed by saying that a is amultiple of ,3, or that ,3 divides a, or that a is a factor or divisor of a.It follows from this definition and Theorem 1 of §13 that we have thefollowing two elementary propositions, mentioned in the Introduction.

1. If a, a' are divisible by a then a + a', a - a' are also divisible byµ;


2. If a' is divisible by a and a is divisible by i then a' is also divisibleby u.

However, it is necessary to pay particular attention to units, that is,to the integers that divide all integers. A unit a must therefore dividethe number 1, and conversely it is evident that every divisor e of thenumber 1 is a unit, because every integer is divisible by 1, and hence(by virtue of proposition 2 above) also by E. At the same time we seethat each product and quotient of two units is itself a unit.

If each of two nonzero integers a, a' is divisible by the other thenwe have a' = ae, where a is a unit. Conversely, if a is a unit theneach of the two integers a and a' = ae is divisible by the other. Wecall two numbers a, a' associates if they are related in this way, andit is clear that any associates of a third number are associates of eachother. In all questions concerning just divisibility, associates behave asa single number. Indeed, if a is divisible by ,3, then every associate of ais divisible by every associate of ,3.

A deeper investigation will enable us to see that two nonzero integersa, 3 have a greatest common divisor, which can be put in the formaa' +,3#' where a' and 3' are integers. This important theorem isnot at all easy to prove with the help of the principles developed thusfar, but we shall later (§30) be able to derive it very simply from thetheory of ideals. I shall therefore end these preliminary considerations ofthe domain of all integers with the remark that there are absolutely nonumbers in this domain with the character of prime numbers. Because,if a is a nonzero integer, and not a unit, then we can decompose it ininfinitely many ways into factors which are integers but not units. Forexample, we have a = v' and also a = /31,32 where )31,,32 are thetwo roots /3 of the equation ,32 - /3 + a = 0. It follows from Theorem 2of §13 that /a--, 31,,32 are integers as well as a.

§ 15. Fields of finite degree

The property of being decomposable in infinitely many ways, whose pres-ence has just been pointed out in the domain of all integers, disappearsagain when the integers under consideration are confined to a field offinite degree. First we have to define the extent and nature of such afield.

Each algebraic number 0, whether an integer or not, evidently satisfies

§ 15. Fields of finite degree 107

an infinity of different equations with rational coefficients, that is, thereis an infinity of polynomial functions F(t) of one variable t which vanishfor t = 9 and whose coefficients are rational. However, among all thesefunctions F(t) there is necessarily one f (t) whose degree n is as smallas possible, and it follows immediately from the well-known method ofpolynomial division that each of the functions F(t) will be algebraicallydivisible by this function f (t), and that f (t) will not be divisible by anypolynomial of lower degree and rational coefficients. For this reason, thefunction f (t) and also the equation f (9) = 0 are called irreducible, andit is clear at the same time that the n numbers 1,01,02'..., 9n-1 forman irreducible system (§4,1).

We now consider the set Il of all numbers w of the form 0(9), where

0(t) = xo + xit + x2t2 + ... + xn_itn-1

is a n y polynomial in t with rational coefficients xo, x1, x2, ... , xn_1 anddegree < n. We first remark that each such number w = 0(9) is uniquelyexpressible in this form, by virtue of the irreducibility of f (t). We thensee easily that the numbers w are closed under rational operations, thatis, addition, subtraction, multiplication and division. For the first twooperations this evidently follows from the common form 0(9) of all thenumbers w, and for multiplication it suffices to remark that each numberof the form '(9), where V)(t) is a polynomial of any degree with rationalcoefficients, is likewise a number w, because if we divide fi(t) by f (t)the remainder will be a function 0(t) of the kind above, and at the sametime we have 0(9) = 0(9). Finally, to treat the case of division it isenough to show that if w = 0(9) is nonzero then its reciprocal w-1 alsobelongs to the system Q. Well, since ¢(t) has no common divisor with theirreducible function f (t), the method for finding the greatest commondivisor of the polynomials f (t), 4(t) gives, as we know, two polynomialsfi (t), 01(t), with rational coefficients, satisfying the identity

f (t)fi (t) + O(t)01(t) = 1.

When t = 9 this gives the result claimed.I call a system A of numbers a (not all zero) a field when the sum,

difference, product and quotient of any two numbers in A also belongsto A. The simplest example of a field is the system of rational numbers.It is easy to see that this field is contained in any other field A since, ifwe choose any nonzero number a in A, it is necessary that the quotient1 of the numbers a and a belong to A, whence the claim follows, since


all the rational numbers are engendered from the number 1 by repeatedadditions, subtractions, multiplications and divisions.

Our system 11 is a field, by the results just proved for the numbersw = 0(0). The rational numbers come from 0(9) by annulling all thecoefficients x1, x2, ... , xn_1 which follow x0. A field 1 obtained froman irreducible equation f (9) = 0 of degree n in the manner indicated iscalled a field of finite degree,t and the number n is called its degree. Sucha field 11 includes n independent numbers, for example 1,0, B2, ... , 9n-1,

whereas any n+1 numbers in the field evidently form a reducible system(§4,1). The latter property, combined with the definition of field, canalso serve as the definition of a field 1 of nth degree. However, I shallnot go into the proof of this assertion.

Now if we arbitrarily choose n numbers

W1=01(0), W2=02(0), ... , Wn = 4n (0)

of the field Il, these numbers (by §4,2) form an irreducible system ifand only if the determinant of their n2 rational coefficients x is nonzero.In this case we call the system of n numbers w1i w2, ... , wn a basis ofthe field Q. Then it is evident that each number w = q5(0) is alwaysexpressible, uniquely, in the form

w = h1w1 + h2w2 + ... + hnWn,

with rational coefficients hl, h2,. . ., hn. And conversely, all numbers wof this form are in Q. The rational coefficients hl, h2, ... , hn will becalled the coordinates of the number w with respect to this basis.

§ 16. Conjugate fieldsWe ordinarily understand substitution to be an act by which objects or el-ements being studied are replaced by corresponding objects or elements,and we say that the old elements are changed, by the substitution, intothe new. Now let 11 be any field. By an isomorphism$ of S2 we mean asubstitution which changes each number

a, 0, a+)3, a-)3, a,3, a/,Q

t If we understand a divisor of a field A to be any field B whose members all belongto A, then a field of finite degree can also be defined as a field with only a finitenumber of divisors. The word divisor (and the word multiple) is used here in asense directly opposite to that used in speaking of modules and ideals, but thisshould not result in any confusion.

t Dedekind calls it a permutation. (Translator's note.)

§16. Conjugate fields

of 1 into a corresponding number

a', Q', (a + Q)', (a -,3)', (a,3)', (a/0)'in such a way that the conditions

109

(1) (a+/3)'=a'+O',(2) (a/3)' = a',Q',

are satisfied and the substitute numbers are not all zero. Weshall see that the set S2' of the latter numbers forms a new field, andthat the isomorphism also satisfies the following conditions:

(3) (a -,3Y = a' -(4) (a/Q)' = a'/,3'

Indeed, if we let a', ,3' be any two numbers in the system 0, then therewill be two numbers a, / in the field 11 changed respectively into a', ,3'.But since the numbers a + 0, a/3 are likewise in S2, it follows from (1)and (2) that the numbers a' + /3', a' f' are in 1'. Thus the numbers ofthe system f' are closed under addition and multiplication. Moreover,since the numbers a = (a -,3) +,3 and a - /3 are likewise in Q, it followsfrom (1) that

a'=(a-/)'+(3',which is condition (3). Thus the numbers of the system 1' are alsoclosed under subtraction. Finally, if ,3' is nonzero then, by virtue of(1), Q will also be nonzero and hence a/f is a number in the field Q.Since we now have a = it follows from (2) that we also havea' = (a//3)'/3', which is condition (4). Thus the numbers of the systemS2' are also closed under division, and consequently S2' is a field. Q.E.D.

We also note that if ,3' = 0 then so too is 3 = 0; otherwise everynumber a in the field S2 could be put in the form (a/0)/3, with the resultthat a' = (a/,3)'/3' = 0, whereas we have agreed that the numbers a'in the system Sl' are not all zero. It evidently follows from this, in viewof (3), that an isomorphism changes two different numbers a, /3 in thefield Il into two different numbers a', /3' in the field Sl', so that eachnumber a' in the field 11' corresponds to a single number a in the fieldQ. The correspondence can therefore be reversed in a unique manner,and the substitution that changes each number a' in the field 1' intothe corresponding number a in the field Sl will be an isomorphism ofthe field 11', because it satisfies the characteristic conditions (1) and (2).


Each of the two isomorphisms will be called the inverse of the other. Wealso call Il and 1' conjugate fields, and any two corresponding numbersa, a' will be called conjugate numbers. For each field SZ there is evidentlyan identity isomorphism of SI, which replaces each number in St by itself.Thus every field is conjugate to itself. It is also easy to see that twofields conjugate to a third are conjugate to each other. Because if eachnumber a in a field SZ is changed by an isomorphism P into a numbera' in a field Il', and a' is likewise changed by an isomorphism P into anumber a" in a field a", then it is clear that the substitution changingeach a in S l into the corresponding a" in S2" is likewise an isomorphismof the field 1, and we denote it by PP'. If we let P-1 denote the inverseisomorphism of P, then PP-1 will be the identity isomorphism of 1,and Sl" will be changed into Il by the isomorphism

(PP')-1 = P/-1P-1

We have previously remarked that each field includes all the rationalnumbers, and it is easy to show that each of these is changed into itselfby an isomorphism of the field. Because, if we take a = ,Q in (4) we get1' = 1, and then, since each rational number can be engendered from 1by a series of rational operations, our proposition follows immediatelyfrom properties (1), (2), (3), (4).

Now let 0 be any number in the field St and let R(t) be any rationalfunction of t with rational coefficients. The number w = R(8), in thecase where the denominator of R(t) does not vanish for t = 0, will alsobe in S2, and if 8 is changed into the number 0' by an isomorphism of thefield, then the number w will be changed into the number w' = R(8'),since it is formed from the number 8, and the rational coefficients of R(t),by rational operations. It follows immediately that if 0 is an algebraicnumber, and hence satisfies an equation of the form 0 = F(8) withrational coefficients, we must also have 0 = F(8'). Thus each number8' conjugate to an algebraic number 8 is likewise an algebraic number,and if 8 is an algebraic integer, 8' will also be an algebraic integer.

After these general considerations, which apply to all fields, we returnto our example of a field l of finite degree n, and pose the problem offinding all the isomorphisms of Q. Since all numbers w in such a fieldare, by §15, of the form 0(e) where 0 is a root of an irreducible nthdegree equation 0 = f (8), it follows from the preceding results that anisomorphism of S2 will be completely determined by the choice of a root0' of the equation 0 = f (0'), because when 0 is changed into 0', w = 0(8)is changed at the same time into w' = 0(0'). Conversely, if we choose

§ 17. Norms and discriminants 111

9' to be any root of the equation 0 = f (9'), and replace each numberw = 0(9) in the field Sl by the corresponding number w' = 0(9'), thenthis substitution will really be an isomorphism of S2, that is, it will satisfyconditions (1) and (2). To show this, we let 01(t), 02(t), ... denote anyparticular functions of the form 0(t). Now if we have

a=01(0), 0=02(0), a+,3=03(0), a,3=04(0),

and consequently

a'=01(0% Y=02(0% (a +,3)' = ()3(9'), (a,3)'=04(0'),

it follows from the equations

03(0)=01(0)+02(0), -04(9) = 01(9)Y'2(9)

and the irreducibility of the function f (t) that we have identically

03(t)=-O1(t)+02(t), 04(t) = O1(t)-02(t) + 05(t)f (t)

Taking t = 9', this gives the equations (1) and (2) we have to show. Ifwe now put

f(t) = (t-9')(t-9")...(t-9(n))then the n roots 01,0 ..... , 9(n) will be different, since the irreduciblepolynomial f (t) cannot have a common divisor with its derivative f'(t),and each of them will correspond to an isomorphism P', P",. .. , p(n) ofthe field Q. The isomorphism P('') changes each number w = 0(9) ofthe field SZ into the conjugate number w(') = O(9(r)) of the conjugatefield SZ(r). To avoid misunderstanding, we point out that the n conju-gate fields S2(''), although derived from Il by n different isomorphisms,nevertheless may not all be different. If they are all the same, SZ willbe called a Galois or normal field.t The algebraic principles of Galoisamount to reducing the study of arbitrary fields of finite degree to thestudy of normal fields; however, lack of space prevents me from goingfurther into this subject.

§ 17. Norms and discriminantsThe norm N(w) of a number w in a field S2 of degree n is the product(1) N(w) = WW,, ... (n)

f Nowadays called a Galois or normal extension (of the rationals); however, it woulddistort Dedekind's meaning to use the term "extension" here. (Translator's note.)


of the n conjugate numbers w', w", ... , w(') of w under the isomorphismsP', pt/,..., P(n). It does not vanish unless w = 0. If w is a rationalnumber, then all the n numbers w(-) equal w, and hence the norm of arational number is its nth power. If a, 0 are any two numbers in thefield I we have (a/3)(-) = a(')#(') and consequently

(2) N(a)3) = N(a)N(O).

The discriminant A(al, a2i ... , an) of any n numbers al, a2, ... , an inthe field Il is the square

(3)// (n) 2A(al, a2, ... , a,,,) ±ala2 ... an )

of the determinant formed by the n2 numbers ai-).By virtue of awell-known proposition in the theory of determinants we then have therelation

(4) 0(1, 0, 02, ... , 0n-1) = (-1)n(n-1)/2N(f'(0)),

and since f'(9) cannot be zero, by the irreducibility of the polynomialf (t), it follows that the discriminant (4) has a nonzero value.

Now if then numbers w1i w2, ... , wn form a basis of the field St (§15),and if

Lo =h1w1+h2w2+"'+hnw,,,

is any number in the field then, since the coordinates hl, h2i .... hn arerational numbers, the isomorphism P(') will change w into the number

w(-) = h1wi-) + h2w2-) + ... + hnwn-),

from which we conclude that

( `5) 0(a1, a2, ... , a.) = a20(W1, w2, ... , W,y),

where a is the determinant of the n2 coordinates of the n numbersal, a2i .,an. This implies, first, that the discriminant of the basisw1, w2, ... , wn cannot vanish, otherwise every discriminant would vanish,contrary to the fact from above that 0(1, 0, 02'...' On-1) is nonzero. Atthe same time it follows that A(a1 i a2, ...,an) will vanish if and onlyif the numbers al, a2, ... , an are dependent on each other (§4,2), andhence do not form a basis of Q.

Since the numbers in a field are closed under multiplication, for anynumber t in St we can put

§ 18. The integers in a field Il of finite degree 113

/2W1 = m1,1W1 + m2,1w2 + + mn,1Wn,

(6) PW2 = m1,2w1 + m2,2W2 + + mn,2Wn,

PWn = m1,nW1 + m2,nW2 +'* * + mn nwn,

where the n2 coordinates mi,i, are rational numbers. Applying the nisomorphisms P(r) yields n2 new numbers of the form

p wir> = ml iW1T) + m2 iw2*) + ... + Mn iW(T)

and, since their determinant is

N(µ) fWIW2 ...W(n) = tml lm2 2 ... mn,n ±W/1W2 ...W(n)

we conclude that LJ

(7) N(p) _ fm1,1m2,2 ... mn,n,

because the determinant

fWr n (n) -1 W2 ... Wn _ A(Wl, W2, ... , Wn)

is nonzero.

It follows that every norm is a rational number and, by virtue of (4)and (5), so is every discriminant. These two propositions could also havebeen deduced from the theory of transformations of symmetric functions,but I wish to avoid relying on this.

If we replace the p in the equations (6) by p - z, where z is any rationalnumber, then the coordinates mi,i, are unchanged except for the mi,i onthe diagonal, each of which is replaced by mi,i - z. Theorem 7 is thenchanged into the equation

m1,1 - z m2,1 ... mn,l

M1,2 m2,2 - z ... mn,2= (µ - z)(µ " - z)

...(µ(n) - z),

I ml,n m2,n ... mn,n - z I

which, being valid for every rational value of z, is necessarily an identityin z. At the same time we see that the n numbers p', p",..., µ(n) con-jugate to the number p are the set of roots of an nth degree equationwhose coefficients are rational numbers.

§ 18. The integers in a field Il of finite degreeAfter these preliminaries, we now pass on to our main objective, thestudy of the integers in the field Il of degree n, the set of which we


denote by o. Since the sum, difference and product of any two integersare also integers (by §13,1), and in 0 (by §15), the domain o, includ-ing all the rational integers, is closed under addition, subtraction andmultiplication. However, the first thing is to put all these numbers in acommon, simple form. The following considerations lead us to it.

Since each algebraic number w is a root of an equation of the form

cwm +C1wm-1 +... +Ctn-lw+Cn = 0,

whose coefficients c, cl, ... , cm_l, cm are rational integers, it follows,multiplying by cl-1, that each such number yields an integer cw whenmultiplied by the nonzero rational integer c. Now if the n numbersw1, w2,. .., wn form a basis of the field S2, we can take nonzero rationalnumbers al, a2i ... , an so that the n numbers

al = alw1, a2 = a2w2, ... , an = anwn

become integers, evidently forming a basis of S2, since they are indepen-dent (by §4,2). It follows that their discriminant 0(al, a2i ... , an) willbe a rational number (by §17), and in fact a nonzero integer since, bydefinition, it is formed from the integers o by addition, subtractionand multiplication. Moreover, we obtain all numbers w in the field S2 byletting the coefficients x1i x2, ... , xn in the expression

run through all rational values. If we restrict their values to be rationalintegers then we certainly obtain only integers w (§13,1). However, it islikely that not all integers in the field S2 will be represented in this way.The situation relates to the following very important theorem:

If there is an integer # of the form

Qklal + k2a2 + ... + knan

k

where k, ki, k2, ... , kn are rational integers without common divisor, thenthere is a basis of the field S2 consisting of n integers ,(31, /32 i ... , Nn whichsatisfy the condition

0(al,a2,...,an) = k2A('31,32...... 3n).

Proof. Since ,Q, al, a2, ... , an are integers, they form a basis of a moduleb = [Q, al, a2i ... , an] which includes only integers from the field Q. Butsince only n of these n + 1 numbers are independent there are (§4,5)n independent numbers 01, /32i ... , Qn which form a basis of the same

§ 18. The integers in a field f2 of finite degree 115

module b = [,31, 132,.. . , ,an] and which are therefore integers in the field.We then have n + 1 equations of the form CO0= cl)3l + C2,32 + ... - COn,

al = C1,101 + c2,1,32 + ... + Cn,la2 = c1,201 + C2,2132 + ... + Cn,2Nn,....................................

an = C1,01 + C2,02 + ... + en,nQn,

whose n(n + 1) coefficients are rational integers and whose n + 1 partialn x n determinants, obtained by suppressing one horizontal line, haveno common divisor (§4,6). If we put

E ±Cl 1C2 2 ... Cn,n = C,

then we have (§17/,(5))

A(al, a2, ... , an) = C2A(Ql, 32, ... ,)3n).

Substituting the preceding expressions for ,Q, al, a2, ... , an in the equa-tion k,Q = k1a1 + k2a2 +... + khan and observing that /31,132,... , Qn

are independent, we find that

kcl = k1C1,1 + k2c1,2 + ... + knCl,n,kc2 = k1c2,1 + k2c2,2 + ... + knc2,n,...................................ken = klcn,l + k2Cn.,2 + ... + knCn n_

If we now replace the elements

Cl,r, C2,r, .... Cn,r

in the rth row of the determinant c by the respective elements

Cl, C2, .. , Cn.,

we conclude from the preceding equations, using a well-known theorem,that the partial determinant obtained has value Sk . Thus the n + 1quantities

ck ck1 ck2 ckn

k, k, k, k

are rational integers with no common divisor, and since the same istrue of the n + 1 numbers k, k1, k2, ... , k n, we necessarily have c = ±k.Q.E.D.

If k > 1, so that the integer ,3 is not in the module a = [al, a2, ... , an],


then there is a basis of the field consisting of n integers Q1, /32,.. . , ,3n withdiscriminant 0(/31, ,32, ... (3n) of absolute value < A(a1, a2, ... , an).Now since the discriminant of every integer basis of the field Q is a ratio-nal nonzero integer, as shown above, there must be a basis w1, W2.... , wn

whose discriminant

A(w1,w2i...,wn) = D

has minimum absolute value. And it follows immediately from the pre-ceding results that, relative to such a basis, each integer

w = hlwl + h2W2 + - .. + hnwn

in the field f must have integer coordinates hl, h2, ... , hn. Moreover, aninteger w is not divisible by a rational integer k unless all its coordinatesare divisible by k. Conversely, since each system of integer coordinateshl, h2i ... , hn produces an integer w, the set o of all integers in the fieldIl is identical with the finitely generated module P1, W2, ... , wn] whosebasis consists of the n independent integers w1i w2, ... , wn.

The discriminant D of such a basis is a fundamentally important in-variant of the field Q. For this reason we call it the fundamental numberor the discriminant of the field 0, and represent it by A(f ). In thesingular case n = 1, SI is the field of rational numbers, and we take itsdiscriminant to be the number +1. To clarify the general case we againconsider n = 2, that is, the case of a quadratic field.

Each root 0 of an irreducible quadratic equation is of the form

0=a+bV,where d is a nonsquare rational integer which is also not divisible bya square (except 1). The numbers a, b are rational, and b is nonzero.The set of all the numbers 0(0) in the corresponding quadratic field isevidently the set of all numbers of the form

w=t+uv/d,where t, u run through all rational values. Under the nonidentity iso-morphism of the field, Vd- changes into -v/d-, and hence w changes intothe conjugate number

w'=t-uvd,which is also in Q. Thus 11 is a normal field (§16). To investigate theintegers w we put

t= x, u= y,

z z

§18. The integers in a field 0 of finite degree 117

where x, y, z are rational integers with no common divisor and z can beassumed positive. Now if w is an integer, so is w' (§16) and consequently

2 2

w+w'=2x, wwx - dyz z2

must also be integers. Conversely, if the latter are integers, then so is w(§13). Suppose that e is the greatest common divisor of z and x. It isnecessary that e2 divide x2 - dy2, hence also dy2 and finally y2, since dis not divisible by any square other than 1. Then e must also divide y,and hence be equal to 1, since x, y, z have no common divisor. Thus zis prime to x while z divides 2x, which means that z = 1 or z = 2. Inthe first case, w = x + yv1d is certainly an integer. In the second case xis odd, so x2 = 1 (mod 4), and since we must have x2 = dy2 (mod 4) itis necessary that y also be odd, and therefore we have d - 1 (mod 4). Ifthe latter condition is not satisfied, that is, if d - 2 or d - 3 (mod 4),then z must equal 1 and so we have o = [1, v/4- and

D=1

v"d2

1 -v =4d.

But if we have d - 1 (mod 4), z can also equal 2,t and we have

1 d 2

O= = 1, 1 2 , and D =1

1 = d.2

These two cases can be combined into one by noting that o = [1, D 2 D ]

in each. It is also clear that a quadratic field is completely determinedby its discriminant D. This is not so for the next case, namely n = 3,where invariants additional to the discriminant are necessary for thecomplete determination of a cubic field. However, we shall be able togive a complete explanation of this fact only with the help of the theoryof ideals.

We now return to considering an arbitrary field Il of degree n, andwe add the following remarks on divisibility and congruence of numbersin the domain o. Let A, p be two such numbers, and suppose that Ais divisible by p. By the general definition of divisibility (§14) we thenhave A = pw, where w is an integer, and since the quotient w of the twonumbers A, p belongs to S2, by definition of a field, w will likewise be anumber in the domain o. The system m of all numbers in the field 0

t It follows, for example, in the case d = -3 that the integers of the field are not allof the form x + y/ where x, y are rational integers.


divisible by it therefore consists of all numbers of the form pw, as w runsthrough all numbers of the domain o = [W1, W2, ... , wn], that is, throughall numbers of the form

w=h1w1+h2w2+"'+hnw,,,,

whose coordinates hl, h2i ... , ham. are rational integers. Consequently wehave m = [pwl, pw2i... , pwn]. We now say that two integers a, /3 in thedomain o are congruent modulo p, and write

a - /3 (mod p),

when the difference a - /3 is divisible by it and hence in m. Thus thiscongruence is completely equivalent to the following:

a - /3 (mod m),

the meaning of which was explained in §2. In the contrary case, a, /3are called incongruent modulo p. If we understand a class modulo p tobe the set of all numbers in o congruent to a particular number, andhence congruent to each other, then, in the notation of §2, the numberof these classes will be (o, m). And since the integers pw1i pw2, ... , pwnforming the basis of m are connected to the numbers w1i w2, ... , wn byn equations of the form (6), (§17), in which the coefficients mz,i, arenecessarily rational integers, it follows from the equation following (7),together with Theorem 4 of §4, that the number of these classes is

(o, m) = ±N(p).

The system m is identical with o if and only if p is a unit, in which case±N(p) = (o, o) = 1.

In this conception of congruence, where an actual number p appearsas divisor or modulus, there is a complete analogy with the theory ofrational numbers. However, it is plain, as we have already indicatedin detail in the Introduction and in Chapter 2, that completely newphenomena concerned with the decomposition of numbers into factorsarise in this same domain o. These phenomena are brought back underthe rule of simple laws by the theory of ideals, the elements of which willbe covered in the next chapter.

4Elements of the theory of ideals

In this chapter we shall develop the theory of ideals to the point indicatedin the Introduction, that is, we shall prove the fundamental laws whichapply to all fields of finite degree, and which regulate and explain thephenomena of divisibility in the domain o of all integers in such a fieldQ. The latter are what we refer to when we speak of "numbers" inwhat follows, unless the contrary is expressly indicated. The theory isfounded on the notion of ideal, whose origin has been mentioned in theIntroduction, and whose importance has been sufficiently illuminated bythe example in Chapter 2 (§§11 and 12). The exposition of the theorythat follows coincides with the one I have given in the second edition ofDirichlet's Vorlesungen fiber Zahlentheorie (§163). It differs mainly inexternal form; however, if the theory has not been shortened it has atleast been simplified a little. In particular, the principal difficulty to besurmounted is now thrown more clearly into relief.

§19. Ideals and their divisibilityAs in the previous chapter, let f be a field of finite degree n, and let obe the domain of integers w in Q. An ideal of this domain o is a systema of numbers a in o with the following two properties:

I. The sum and difference of any two numbers in a also belong to a,that is, a is a module.

II. The product aw of any number a in a with a number w in o is anumber in a.

We begin by mentioning an important special case of the concept ofideal. Let y be a particular number; then the system a of all numbersa = pw divisible by p forms an ideal. We call such an ideal a principal

119

120 Chapter 4. Elements of the theory of ideals

ideal and denote it by o(p), or more simply by op or po. It is evidentthat this ideal will be unchanged when p is replaced by an associate,that is, a number of the form ep, where a is a unit. If p is itself a unit wehave op = o, since all numbers in o are divisible by p. It is easy to seethat no other ideal can contain a unit. Because if the unit a is in the ideala then (by II) all products ew, and hence all numbers w in the principalideal o, are in a. But since, by definition, all numbers in the ideal a arelikewise in o, we have a = o. The ideal o plays the same role among theideals as the number 1 plays among the rational integers. The notionof principal ideal op also includes the singular case where p = 0, wherethe resulting ideal consists of the single number zero. However, we shallexclude this case from now on.

In the case n = 1, where our theory becomes the old theory of num-bers, every ideal is evidently a principal ideal, that is, a module of theform [m] where m is a rational integer (§§1 and 5). The same is truefor the special quadratic fields considered in Chapter 2 (§6 and the be-ginning of §7). In all these cases, where every ideal of the field Il isa principal ideal, numbers are governed by the same laws that governthe theory of rational integers, because every indecomposable numberalso has the character of a prime number (see the Introduction and §7).This will follow easily from the results below, but I mention it now toencourage the reader to make continual comparisons with the specialcases, and especially with the old theory of rational numbers, becausewithout doubt it will help greatly in understanding our general theory.

Since each ideal is a module (by virtue of I), we immediately carryover to ideals the notion of divisibility of modules (§1). We say that anideal m is divisible by an ideal a, or that it is a multiple of a, when allthe numbers in m are also in a. At the same time we say that a is adivisor of in. According to this definition, each ideal is divisible by theideal o. If a is a number in the ideal a then the principal ideal oa willbe divisible by a (by II). For this reason we say that the number a, andhence every number in a, is divisible by the ideal a.

Likewise we say that an ideal a is divisible by the number 77 when ais divisible by the principal ideal o77. When all the numbers a in anideal a are of the form 77P it is easy to see that the system r of all thenumbers p = a/77 will form an ideal. Conversely, if p runs through allthe numbers in an ideal r while 77 is a fixed nonzero number, then all theproducts 77P will again form an ideal, divisible by or7. We shall denotean ideal formed in this way from an ideal r and number 77 by rq or 77r.We evidently have (rr7)r7' = r(7777') = (r77')r7, and 77r' will be divisible by

§20. Norms 121

r7r if and only if r' is divisible by r. Thus the equation 77r' = it entailsthe equation r' = r. The notion of a principal ideal op is the special caseof rp where r = o.

We finally remark that divisibility of the principal ideal op by theprincipal ideal or7 is completely equivalent to divisibility of the numberp by the number i . The laws of divisibility of numbers in o are thereforeincluded in the laws of divisibility of ideals.

The least common multiple m and the greatest common divisor a oftwo ideals a, b are also ideals. Certainly m and a are modules (§1, 3and 4), and they are divisible by o, since a and b are divisible by o.Also, if p = a = 3 is a number in m and hence also in a and b, and if6 = a' +/3' is a number in the module a then the product pw = aw = /3wwill likewise be in m and the product 6w = a'w +,a'w will be in a since(by virtue of II) the products aw, a'w are in a and the products /3w, /3'ware in b. Thus m and a enjoy all the properties of ideals. At the sametime it is clear that mr7 will be the least common multiple of the idealsall, bq, and ar7 their greatest common divisor.

If b is a principal ideal o77, then the least common multiple m of a, bwill always be of the form r7r, where r is another ideal and in fact a divisorof a, since 77a is a common multiple of a and o77, and hence divisible by77r. This case occurs frequently in what follows, and for that reason wesay, for brevity, that the ideal r dividing the ideal a corresponds to thenumber 77. If r' is the divisor of r corresponding to the number 77', thenr' will also be the divisor of a corresponding to the product q?7'. Thisis because 7rr7'r' is the least common multiple of 77r and or777', and hencealso of a and or777', since 77r is the least common multiple of a and o77,and or777' is divisible by or7.

§20. NormsSince each ideal a is also a module, we say that two numbers w, w' inthe domain o are congruent or incongruent modulo a according as theirdifference w - w' belongs to a or not. We express the congruence of wand w' modulo a (§2) by the notation

w - W (mod a).

As well as the theorems on congruences established previously for ar-bitrary modules, we must also note that two congruences modulo the


same ideal a,

W=-W, 'w" -W "' (mod a),

can be multiplied to give the congruence

ww" - w' "w' (mod a),

since the products (w-w')w" and (w"-and hence also their sumww" - w'ware numbers in the ideal a. Moreover, if m is a principalideal op, then (by virtue of §18) the congruence w - w' (mod m) will beidentical with the congruence w - w' (mod p).

A very important consideration is the number of different classes, mod-ulo the ideal a, which make up the domain o. If p is a particular nonzeronumber in the ideal a then the principal ideal op will be divisible by a,and since a is divisible by o it follows (§2,4) that

(o, op) _ (o%a)(a, op).

But the number (o, op) = ±N(p) by §18, and hence the domain o con-tains only a finite number of mutually incongruent numbers modulo theideal a (§2,2). This number (o, a) will be called the norm of the ideal aand we denote it by N(a). The norm of the principal ideal op is equalto ±N(p), and o is evidently the only ideal with norm 1.

If p runs through a complete system of N(a) incongruent numbers(mod a) then so does (1 + p), and adding the corresponding congruences1 + p = p', where p' runs through the same values as p, yields N(a) = 0(mod a). That is, N(a) is always divisible by a. As a special case, thisresult includes the evident theorem that N(p) is divisible by p (see §17).

Now suppose that r is any ideal and 77 is a nonzero number. Always,

(o77, ri7) = (o, r) = N(r),

since two numbers i'w' and 77w" in the principal ideal 77o are congruent(mod 77r) if and only if the numbers w', w" in o are congruent (mod t).

Let a, b be any two ideals, let m be their least common multiple, andlet a be their greatest common divisor. By §2, 3 and 4, we have

(b, a) m) a)

and, since a is divisible by o,

(o, a) = (o, D) (D, a), (o, m) _ (o, b) (b, m),

hence

N(a) = (b, a)N(a), N(m) = (b, a)N(b)

§21. Prime ideals 123

and

N(m)N(0) = N(a)N(b).

If we apply these theorems to the case where b is a principal ideal oi',so that m is of the form rrl, then, since the ideal r is the divisor of acorresponding to the number i (§19), we get

(b, a) _ (oi', rrl) = N(r),

and consequently

N(a) = N(r)N(D).

The ideal r can also be defined as the system of all roots p of the con-gruence q7P - 0 (mod a), as is easy to see.

§21. Prime idealsAn ideal p is called prime when it is different from o and divisible by noideals except o and p. This definition yields the following theorems:

1. Each ideal a different from o is divisible by a prime ideal.

This is because the ideals different from o that divide the ideal ainclude one, p, whose norm is smallest, and the latter is certainly aprime ideal. Indeed, if 0 is an ideal dividing p, but different from p ando, then we have ('0,p) > 1. Hence N(p) = (a, p)N(0) > N(0) and awill be a divisor of the ideal a, different from o and with norm < N(p),contrary to hypothesis. Thus p is a prime ideal. Q.E.D.

2. If the number rl is not divisible by the prime ideal p, then qp willbe the least common multiple of the two ideals p and orl.

The least common multiple of p and orl is in any case of the form rlr,with the ideal r a divisor of ap and hence equal to o or p. But r cannotbe o, because qo is not divisible by p; consequently r = p. Q.E.D.

3. If neither of the two numbers rl, p is divisible by the prime ideal p,then their product q7P will not be divisible by p.

Otherwise the ideal 77(op) would be a common multiple of p and oil.Consequently it would be divisible by the least common multiple IN of pand orl. But the divisibility of 77(op) by ilp implies (§19) the divisibilityof op by p, contrary to hypothesis. Thus ?7A is not divisible by p. Q.E.D.


It follows immediately from this that all the rational numbers divisibleby a prime ideal p, amongst which we have the number N(p) (§20), forma module [p], where p is a positive rational prime number. This is becausethe smallest positive rational p divisible by p cannot be a compositenumber ab, otherwise one of the smaller numbers a, b would likewise bedivisible by p, and p cannot be 1 otherwise we should have p = o (§19).And each rational integer m divisible by p must be divisible by p, asbecomes evident by putting m in the form pq + r, since the remainderr = m - pq is also divisible by p. Now with op being divisible by p, andhence N(op) = p"` being divisible by N(p) (§20), N(p) will be a powerpf of p, and the exponent f will be called the degree of the prime idealp.

4. If the ideal a is divisible by the prime ideal p then there is a number77 such that r7p is the least common multiple of a and o77.

This important theorem is evident when we have a = p, because anynumber 77 not divisible by p, for example r7 = 1, satisfies the condition.But if a is different from p we first confine ourselves to showing the exis-tence of a number 77 such that the divisor r of the ideal a correspondingto 77 is at the same time divisible by p, while having norm less than thatof a. Since we have N(a) = N(r)N(D), where a is the greatest commondivisor of a and oq (§20), the latter condition depends on choosing 7) sothat N(a) > 1, which means that a is different from o. To attain thisgoal, and at the same time make r divisible by p, we distinguish twocases:

First, if all the ideals (except o) that divide a are divisible by p, thenwe choose 77 to be a number divisible by p but not divisible by a, whichis always possible because p is not divisible by a. Then it is clear thata will be divisible by p, and hence will be different from o. Moreover,since 77 is not divisible by a, and q r is, r will also be different from o,and hence divisible by p.

Second, if there is an ideal e different from o which divides a and isnot divisible by p, then we choose q to be a number divisible by e butnot divisible by p. Then a will be divisible by e and hence again differentfrom o. Moreover, since 77r is divisible by a and hence also by p, r willalso be divisible by p, because 77 is not divisible by p (by 1).

Having established the existence in both cases of at least one numberr7 with the required property, we see without difficulty that we certainlyhave r = p if we choose 77 so that N(r) is as small as possible. Because,if the ideal r divisible by p is not equal to p, we can proceed with r the

§22. Multiplication of ideals 125

same way as with a, and choose a number 77' in such a way that thedivisor r' of r corresponding to r7' again has norm less than that of r,while likewise being divisible by p. But since (§19) r' is at the same timethe divisor of a corresponding to the number 77r7' this contradicts theassumption we have made about r7 and r. Thus r = p, that is, q p is theleast common multiple of a and or7. Q.E.D.

§22. Multiplication of idealsIf a runs through all the numbers in an ideal a, and /3 runs throughthose of an ideal b, then all the products of the form a3, together withtheir sums, form an ideal c. These numbers are in o and they are closedunder addition; also under subtraction, because the numbers (-a) arelikewise in a. Finally, each product of a number Ea,/3 in c by a numberw in o also belongs to c, since each product aw again belongs to a. Thisideal c is called the product of the two factors a, b, and we denote it byab.

It follows immediately from this definition that oa = a, ab = ba and,if c is any third ideal, (ab)c = a(bc), whence we conclude by well-knownargumentst that in a product of any number of ideals al, a2,. - ., a..the order of the multiplications, which combine two ideals into a sin-gle product, has no influence on the final result, which can be writtensimply as al a2 a,,, and evidently consists of all numbers of the formEala2 ... am, where al, a2, ... , a,. are numbers from the respective fac-tors al, a2,. .., a,,,. If all the m factors equal a their product will becalled the mth power of a, and we denote it by a. We also put ao = o,al = a and in general we have aras = ar+s, (ar)s = ars. In addition, weevidently have a(oi) = a77 and (or7)(oi7') = or777'. Finally, we re-establishthe following theorems:

1. The product ab is divisible by the factors a and b.This is because (by virtue of property II) each product aw, hence each

product a,3, and hence (by I) each sum of such products belongs to a.That is, ab is divisible by a.

2. If a is divisible by a', and b by b', then ab is divisible by a'b'.This is because all the numbers Eal in ab are in a'b', since a is in a

and hence in a', and ,Q is in b and hence in W.

t See §2 of Dirichlet's Vorlesungen fiber Zahlentheorie.


3. If neither of the ideals a, b is divisible by the prime ideal p, thenthe product ab is also not divisible by p.

This is because there are numbers a, 0, in a, b respectively, which arenot divisible by p, and then the number aj3 in ab is also not divisible byp (§21,3).

§23. The difficulty in the theoryIt would be easy to augment considerably the number of theorems con-necting the notions of divisibility and multiplication of ideals, and wemention without proof the following propositions, simply to emphasizethe resemblance with the corresponding propositions in the theory ofrational numbers.

If a, b are relatively prime ideals, that is, having greatest commondivisor o, then their least common multiple is ab, and at the same time

N(ab) = N(a)N(b).

If p is a prime ideal, and a is any ideal, then either a is divisible by por a and p are relatively prime.

If a is an ideal relatively prime to b and c, then a is also relativelyprime to bc.

If ab is divisible by c and a is relatively prime to c, then b is divisibleby c.

However, all these propositions are insufficient to complete the anal-ogy with the theory of rational numbers. It is necessary to rememberthat divisibility of an ideal c by an ideal a, according to our definition(§19), means only that all the numbers in the ideal c are also in theideal a. It is very easy to see (§22,1) that any product of a by an idealb is divisible by a, but it is by no means easy to show the converse,that each ideal divisible by a is the product of a by an ideal b. Thisdifficulty, which is the greatest and really the only one presented by thetheory, cannot be surmounted by the methods we have employed thusfar, and it is necessary to examine more closely the reason for this phe-nomenon, because it is connected with a very important generalisationof the theory. By attentively considering the theory developed until now,one notices that all the definitions retain their meaning, and the proofsof all theorems still hold, when one no longer supposes that the domaino consists of all integers in the field Q. The only properties of o reallyneeded are the following:

§23. The difficulty in the theory 127

(a) The system o is a finitely generated module [w1, w2, ... , wn] whosebasis is also a basis for the field Q.

(b) The number 1 is in o, hence so are all the rational integers.(c) Each product of two numbers in o is also in o.When a domain o enjoys these three properties we shall call it an

order. It follows immediately from (a) and (c) that an order consistsentirely of integers from the field Sl, but it does not necessarily containall the integers (except in the case n = 1). Now if a number a in theorder o is called divisible by a second such number p only when a = µw,with w also in o, and if we modify the notion of congruence of numbers ino in the same manner, then one sees immediately that the number (o, op)of mutually incongruent numbers of o modulo p is again ±N(µ) (§18).It is also easy to see that all the definitions and all the theorems of thepresent chapter retain their meaning and truth if we always understandnumber to mean a number in the order o. Thus, in particular, each ordero in field 1 has its own theory of ideals, and this theory is the same forall orders (which are infinite in number) up to the point we have carriedit so far. However, while the theory of ideals in the order o of all integersin the field 1 leads finally to general laws which coincide completely withthe laws of divisibility for the rational numbers, the theory of ideals inother orders is subject to certain exceptions, or rather, it requires acertain limitation of the notion of ideal. The general theory of ideals inan arbitrary order, whose development is equally indispensable for thetheory of numbers and which, in the case n = 2, coincides with the theoryof orders of binary quadratic forms,t will be left aside in what follows,tand I shall content myself with giving an example to call attention tothe character of the exceptions just mentioned. In the quadratic fieldresulting from a root

+vf ---32

of the equation 92 + 9 + 1 = 0 the module [1, VI '--3] forms an order o notcontaining all the integers of the field. The modules [2,1+v/-3] = p and[2, 2vl---'-3] = 0(2) must be regarded as ideals in this order o, inasmuch asthey enjoy the properties I and II (§19). However, while o(2) is divisibleby p, there is no ideal q in o such that pq = o(2).

t Disquisitiones Arithmeticae, art. 226.$ I treat this theory in detail in the recently published memoir: "Ueber die An-

zahl der Ideal-Classen in den verschiedenen Ordnungen eines endlichen Korpers."(Festschrift zur Sdcularfeier des Geburtstages von C.-F. Gauss. Braunschweig, 30April 1877).


§24. Auxiliary propositionsTo achieve the completion of the theory of ideals in an order o containingall the integers of a field 1 we need the following lemmas, which are nottrue unless we restrict ourselves to such a domain o.

1. Let w, p, v be three nonzero numbers in o such that v is notdivisible by u. Then the terms of the geometric progression

(AV)2,

w

(µ)3,

up to a termCvle

W

µ/

at some finite position, will all be in o, and beyond that none of themwill be an integer.

In fact, if the number of integral terms exceeds the absolute value kof N(w) then there must be (by §18) at least two among them, corre-sponding to exponents s and r > s, which are congruent modulo w. Butsuch a congruence,

LO C-)r w ( )3 )(mod w),

implies that the numberv

µin the field S2 satisfies an rth degree equation of the form

7]r = 77s + W

where w' is an integer, and hence (§13,2) 7) itself is an integer, contraryto our hypothesis that v is not divisible by p. Thus at most k termsof the series can be integers, and hence members of o. Moreover, if theterm

P-W (AV)r,

with r > 1, is an integer, and if s is any one of the r exponents0, 1, 2, ... , r - 1 then the term

=_

W C(v

s

µ)


will also be an integer, because

0.r = wr-sps

is an integer (§13,2). This completes the proof of the proposition.

2. Let p, v be two nonzero numbers in o, with v not divisible by A.Then there are two nonzero numbers rc, A in o such that

K V

A p

and s;2 is not divisible by A.

If

are the last two integral terms of the series2 3

V

v vJul

/I (A) , A t' (A)and hence in o, then we evidently have e > 1 and

r6 V

A µ

Thus n2 is not divisible by A.

/C2 v

A =µ (A

e+1

Q.E.D.

§25. Laws of divisibilityWith the aid of these lemmas it is easy to bring the theory of ideals inthe domain o to the desired conclusion, which is found in the followinglaws:

1. If p is a prime ideal then there is a number A divisible by p, and anumber r. not divisible by p, such that rcp is the least common multipleof oA and or..

Proof. Let p be any nonzero number in the prime ideal p. Since opis divisible by p there is a number v such that vp is the least commonmultiple of op and ov (§21,4). This number v cannot be divisible by p,otherwise the least common multiple of op and ov would be ov and notpv. Now if we choose (§24,2) the two numbers K, A so that rcp = Avand rc2 is not divisible by A, then (§19) the ideal rcvp will be the least


common multiple of c(op) = oAv and oicv. It follows (§19) that icp isthe least common multiple of oA and oK. Then p is the divisor of theprincipal ideal oA corresponding to the number #c. However, is is notdivisible by p, otherwise 0c2 would be divisible by rcp and hence also byA.

2. Each prime ideal p has a multiple, by an ideal a, which is a principalideal.

Proof. We retain K and A from above with the same meaning, andlet a be the greatest common divisor of oA and or.. We shall showthat pa = oA. In fact, since all numbers in the ideal a are of the formb = rcw + Aw', where w, w' are two numbers from o, if w is any numberin p we have wS = icww + Aww' - 0 (mod A), because Kp and hencealso Kw is divisible by oA. Thus pa is divisible by oA. Conversely, if Kis not divisible by p, in which case o is the greatest common divisor ofor. and p, we can express the number 1 in o as Kw + w, with w in p andza in p. We then have A = A . Kw + w A m 0 (mod po), because thefirst factors A, zo are in p and the second factors Kcw, A are in Z. Thuseach of the two ideals pa, oA is divisible by the other, and consequentlypa = oA. Q.E.D.

3. If the ideal a is divisible by the prime ideal p then there is exactlyone ideal a' such that pa' = a, and at the same time N(a') < N(a).

Proof. Suppose, as we now can, that pa = oA. Then if a is divisibleby p, and hence as by pa (§22,2), we have as = Aa' where a' is an ideal(§19). Multiplying by p, we get Aa = Apa' and hence also a = pa'. Nowlet b be an ideal also satisfying the condition pb = a. Multiplying theequation pb = pa' by a we get Ab = Aa', whence b = a'. In addition,there is (§21,4) a number 77 such that 77p is the least common multiple ofa and oil. But with 77p being divisible by a = a'p it follows, multiplyingby a, that o77A is divisible by Aa', and hence that 77 is divisible by a'.However, q is certainly not divisible by a, otherwise oil and not qp wouldbe the least common multiple of a and or7. Thus, since 77 is divisible bya' and not by a, it follows that a' is different from a, and consequentlywe have N(a') < N(a), because a' is a divisor of a. Q.E.D.

4. Each ideal a different from o is either a prime ideal or else express-ible as a product of prime ideals, and in only one way.

Proof. Since a is different from o there is (§21,1) a prime ideal pidividing a, and hence we can put (by 3) a = plat, where N(al) < N(a).


If we have al = o then a = pl will be a prime ideal. But if N(al) is > 1,so that a1 is different from o, we can likewise put a1 = p2a2 where P2 isa prime ideal and N(a2) < N(al). If N(a2) > 1 we can continue in thesame manner, obtaining ideals al, a2i a3, ..., whose norms are smallerand smaller, finally arriving at the ideal o = am after a finite number ofdecompositions. We then have a product of m prime ideals

a = p1p2...pm,

Now if at the same time

a = g1g2 ... qm,

where q1, q2, ... , qm are also prime ideals, then q1 will be a divisor ofthe product P1P2 pm and hence (§22,3) at least one of its factors, pifor example, must be divisible by q1. And since pi is divisible only bythe two ideals o and pi, we necessarily have ql = pi, since q1 is differentfrom o. We then have

p1(p2p3 ... pm) = pl (g2q3 " ' qm),

whence (by 3)

P2P3 ... Pm = g2q3... qm,

We can continue in the same manner, just as in the theory of rationalnumbers,t and thus arrive at the result that each prime ideal factor inone product occurs in the other, and exactly the same number of times.Q.E.D.

5. Each ideal a, when multiplied by a suitable ideal m, becomes aprincipal ideal.

Proof. Suppose that a = plp2 pm. By 2, we can multiply the primeideals p1, P2, ... , pm by the corresponding ideals 01, 02i ... , am to changethem into principal ideals plat, p2a2i ... , If we now put

m = 002 ... am,

then am = (P101)(p2a2) will be a unique product of principalideals, and hence itself a principal ideal. Q.E.D.

6. If the ideal c is divisible by the ideal a then there is an ideal b, andonly one, satisfying the condition ab = c. If the product ab is divisibleby the product ab' then b will be divisible by b', and ab = ab' impliesb=b'.t See Dirichlet's Vorlesungen fiber Zahlentheorie, §8.


Proof. We choose the ideal m so that am is a principal ideal op. Nowif c is divisible by a, and hence cm by am (§22,2), we can, by §19, putcm = µb, where b is an ideal. Multiplying by a, we get µc = pab, whencec = ab. Now let a, b, b' be any ideals, and suppose that ab is divisibleby ab'. Again it follows, multiplying by m (§22,2), that µb is divisibleby pb', and hence (§19) that b is divisible by W. If in addition ab = ab',then each of the ideals b, b' will be divisible by the other, that is, b = W.Q.E.D.

7. The norm of a product of ideals is equal to the product of thenorms of the factors: N(ab) = N(a)N(b).

Proof. We first consider the case of a product a = pa', where the factorp is a prime ideal. Since a is divisible by p, there is (by 3) a number rldivisible by a' but not by a, and r/p is the least common multiple of aand orl. Then we have (§20) N(a) = N(p)N(D), where a is the greatestcommon divisor of the ideals a and ord. Since a and of are divisible by a',a must also be divisible by a' (§1,4), and hence there is (by 6) an ideal nsatisfying the condition na' = Z. Moreover, since a is divisible by a, pa'is divisible by na', so the prime ideal p must (by 6) be divisible by n, andthus we must have n = p or o. The first equation is impossible unlesswe have a = pa' = a, so that rl is divisible by a, and this is not the case.We therefore have n = o, whence a = a' and also N(pa) = N(p)N(a'),which proves the theorem in the case considered. However, the generaltheorem is an immediate consequence. Since each ideal (other than o)is (by 4) of the form

a = p1p2...pm,

where pl, p2, ... , p. are prime ideals, it follows that

N(a) = N(pi)N(p2p3...pm) = N(pl)N(p2)N(p3...pm)

and hence

N(a) = N(pi)N(p2) ... N(pm)

Moreover, if we have

b = gig2...qr,

where qi, q2, , qr again denote prime ideals, then we get

ab = p1p2 ... pmglg2 ... qr,


and consequently

N(b) = N(gl)N(g2) ... N(q,),N(ab) = N(pi) ... N(pm)N(ql) ... N(qr),

which obviously implies

N(ab) = N(a)N(b).

Q.E.D.

8. An ideal a (or a number a) is divisible by an ideal a (or a number6) if and only if each power of a prime ideal which divides a (or 6) alsodivides a (or a).

Proof. If p is a prime ideal and p7z is a divisor of an ideal a then wehave (by 6) a = alp"°, where al is an ideal. If we suppose the latterdecomposed into all its prime factors, then a will also be expressed asa product of prime ideals, among which the factor p appears at least mtimes. Conversely, if the decomposition of Z into prime factors includesthe prime ideal p at least m times, then a will evidently be divisibleby pm. Thus if we suppose that every power of the prime ideal whichdivides a also divides an ideal a, this amounts to saying that all theprime factors in the decomposition of a appear, at least as often, asfactors in the decomposition of a. The factors of a include first of allthe factors of a and, if we denote the product of the other factors by a',then we have a = aa', and consequently a is divisible by D. The converseproposition, that if a is a divisor of a then each power of a prime idealthat divides a also divides a, is verified easily. Q.E.D.

If we combine all factors of the same prime in the decomposition ofan ideal a then we find

a = pagbrc . .

where p, q, r, ... are different prime ideals. By virtue of the precedingtheorems we can show that all the divisors a of a are given by the formula

a = pa'gb'rc, ..

where the exponents a', b', c', ... satisfy the conditions

0<a'<a, O<b'<b, 0<c'<c, ....Since any two different combinations of exponents a', b', c', ... corre-spond (by 4) to different ideals a, the total number of different divisorswill be (a+ 1)(b + 1)(c + 1)...

134 Chapter .4. Elements of the theory of ideals

9. If a is the greatest common divisor of the two ideals a, b then wehave

a=aa', b=ab',

where a', b' are relatively prime ideals, and the least common multiplem of a, b is Da'b' = ab' = ba'. Moreover, if ae is divisible by b, e will bedivisible by W.

We leave the task of finding the proof of this proposition to the reader,along with the rules that allow m, a to be derived from the decomposi-tion of a, b into prime factors.

§26. CongruencesHaving established the laws of divisibility for ideals, and hence also fornumbers in o, we shall add some considerations on congruences which areimportant for the theory of ideals. For the moment we shall be contentsimply to give indications of the proofs.

1. Since o is the greatest common divisor of two relatively prime idealsa, b, and ab is their least common multiple, then (§2,5) the system oftwo congruences

w - p (mod a), w - or (mod b),

where p, or are two given numbers in o, always has roots w, and all theseroots come under the form

w - T (mod ab),

where r represents a class of numbers modulo ab which is determined byp and o, or by their corresponding classes modulo a and b respectively.Conversely, each class r (mod ab) is determined in this way by preciselyone combination p (mod a), Q (mod b).

We now say that the number p is prime to the ideal a when op and a arerelatively prime ideals, and we let b(a) denote the number of mutuallyincongruent numbers modulo a which are prime to a. One easily derivesthe theorem that

1'(ab) = V)(a)V)(b)

for two relatively prime ideals a, b since r is prime to ab if and only if pis prime to a and or is prime to b. Thus we only need to find the value of


the function vi(a) in the case where a is a power pm of the prime idealp. The total number of mutually incongruent numbers modulo p7z is, inthe case m > 0, equal to

N(pm) = [N(p))m = (o, pm) = (o, p) (p, pm) = (p, pm)N(p)

It is necessary to subtract from this the number of numbers not primeto p"°, and hence divisible by p. Since this number is equal to

(p, pm) = [N(p)]m-1,

we get

'0(pm) = [N(p)]m - [N(p)]m-1 = N(pm) l 1- N(p) /whence we conclude immediately, by virtue of the preceding theorem,that

V(a) = N(a) 1 - Nl )(p)/

where the product fl is taken over all the distinct prime ideals p thatdivide the ideal a. Since we also have

'i(o) = 1,

we arrive, just as in the theory of rational numbers,t at the theorem

O(d) = N(a),

where the summation is taken over all the ideals a' that divide a.

2. If D is the greatest common divisor of the ideals a and orb then wehave a = Da', and r/a' will be the least common multiple of a and oi',that is, a' will be the divisor of a corresponding to the number i (§19).Conversely, if 77a' is the least common multiple of a and oil then we havea = Da', where D is the greatest common divisor of a and orj. It is alsoclear that the complementary factors D and a' of the ideal a remain thesame for all numbers congruent to 77 modulo a. It is evident that it willalso be the same if we replace 77 by a number i' - iiw (mod a) where wis a number prime to a'. And conversely, if the greatest common divisorD of a, o?7 is also that of a, oi' then

7 rlw, r/ r/'w' (mod a),

t See Dirichlet's Vorlesungen fiber Zahlentheorie, §14.


whence we deduce

77ww' - 77 (mod a), ww' - 1 (mod a'),

and consequently w is a number prime to a'. Thus the number of mu-tually incongruent numbers 77 modulo a which correspond to the samedivisor a' of a is '(a'). However, it is necessary to point out that herewe have assumed the existence of at least one such number 77. Thus ifwe are given an arbitrary divisor a' of the ideal a the most we can affirmso far is that the number X(a') of mutually incongruent numbers 77 mod-ulo a which correspond to the same divisor a' will equal either 'O(d) orzero. To decide this alternative we consider all the incongruent numbersmodulo a, which are N(a) in number, and partition them into groups ofX(d) numbers, corresponding to the divisors a'. We must have

X(d) = N(a)

where the summation is taken over the divisors a' of a. But since wealso have (1)

E O(d) = N(a),

it follows immediately that X(d) is never zero and always 0(a'). Thuswe have proved the following very important theorem:

"If a and a' are any two ideals we can always, by multiplying a by anideal b' prime to a', change it into a principal ideal OW = 077. ))

Putting aa' = a, the fact that z'(a') is nonzero means that there isalways a number 77, corresponding to the divisor a' of a, such that a willbe the greatest common divisor of a and o17. If we then put o77 = ab', b'will be an ideal prime to a'. Q.E.D.

3. Since each product pp' of numbers p, p' prime to an ideal a islikewise a number prime to a, and since, as p remains constant and p'varies, pp' runs through a system of /1'(a) mutually incongruent numbers(mod a), we deduce by the well-known method,t and for each value ofthe number p, the congruence

pO(a) = 1 (mod a),

which represents the highest generalisation of a celebrated theorem of

t See Dirichlet's Vorlesungen fiber Zahlentheorie, §19.


Fermat. For a prime ideal p we conclude easily that every number w inthe domain o satisfies the congruence

wN(p) = w (mod p),

that is, the congruencewPf = w (mod p),

where p is the rational prime number divisible by p and f is the degreeof the prime ideal p (§21,3). This theorem is as important for the theoryof the domain o as the theorem of Fermat is for the theory of rationalnumbers, as we shall at least try to make clear by the following remarks,space not permitting us to pursue the general theory further.

If the coefficients of the polynomial function F(x), of degree m, arein o, and if the coefficient of the highest degree term is not divisible bythe prime ideal p then we deduce, by well-known reasoning,t that thecongruence F(w) = 0 (mod p) cannot have more than m mutually incon-gruent roots. This proposition, combined with the preceding theorem,leads to a complete theory of binomial congruence modulo p. We de-duce, among other things, the existence of primitive roots for the primeideal p, meaning numbers ry whose powers

1, 'y, ''2, ... , lyN(p)-2

are mutually incongruent. In general, the theory of congruences of higherdegree with rational coefficients carries over completely to functions F(x)whose coefficients are numbers in the domain o.

However, we have previously ascertained a close connection betweenthe theory of ideals and the theory of higher degree congruences, re-stricted to the case of rational coefficients, developed in the works ofGauss, Galois, Schonemann and Serret.t Since all ideals are composedof prime ideals, and each prime ideal p divides a determinate rationalprime p, we obtain a complete overview of all the ideals in the domaino by decomposing all ideals of the form op into their prime factors. Thetheory of congruences provides a procedure capable of doing this in agreat number of cases. If 0 is an integer of the field SZ and

A(1, 0, e2, ... , 9n,-1) = k2A(ul)

then, if p is not a divisor of k we decompose op into prime ideals as

t See Dirichlet's Vorlesungen fiber Zahlentheorie, §26.t See my memoir Abriss einer Theorie der hoheren Congruenzen in Bezug auf einen

reellen Primzahl-Modulus (Crelle's Journal, 54).


follows. If f (t) is the polynomial of degree n in t that vanishes for t = 0we can put

f (t) = Pl(t)a1P2(t)a2 ... Pe(t)ae (mod p),

where P1 (t), P2 (t), ... , Pe (t) are distinct irreducible polynomials withrespective degrees fl, f2i ... , fe. Then we certainly have

op = p1 p22 ... pe

where p1, p2, ... , Pe are distinct prime ideals with respective degreesfl, f2, .... fe. We deduce from this the following extremely importanttheorem:

"The rational prime p divides the fundamental number 0(1l) of thefield S2 if and only if p is divisible by the square of a prime ideal."

This theorem is still true, although the proof is more difficult, whenthe numbers k corresponding to all possible numbers 0 are all divisibleby p. Such a case is actually encountered,t and this is one of the reasonsI was determined to build the theory of ideals, not on congruences ofhigher degree, but on entirely new principles which are at the same timeas simple as possible and better suited to the true nature of the subject.

§27. Examples borrowed from circle divisionBy the general theory of ideals, the fundamentals of which I have devel-oped above, the phenomena of divisibility of numbers in each domain o,consisting of the integers of a field SZ of finite degree, have been reducedto the same fixed laws that govern the old theory of rational numbers.If we reflect on the infinite variety of these fields 1, each of which hasits own theory of special numbers, it is undoubtedly within the spirit ofgeometry to ascertain those general laws obeyed by the various theorieswithout exception. However, this is not only of aesthetic or purely the-oretical interest, it is also of practical value. The knowledge that thesegeneral laws exist greatly assists in the discovery and proof of specialphenomena in a given field Q. To back up this claim to the full wewould admittedly need to develop further the general theory of ideals,and to combine it in particular with the algebraic principles of Galois.Instead, I shall simply try to show, using the example that led Kummer

t See the Gottingische gelehrte Anzeigen of 20 September 1871, p. 1490.


to introduce his ideal numbers in the first place, how the elements of thetheory explained above lead to the goal with great facility.

Let m be a positive rational prime number, and let Il be the field ofdegree n resulting, in the manner described above (§15), from a primitiveroot of the equation 9' = 1, that is, a root of the equation

f(o)=om-1+em-2+...+92+e+1=0.Since the coefficients are rational, we have n < m - 1. Moreover, since0,02'. .., 0--1 are all roots of this equation we have

.f(t) = tt1 = (t - 9)(t - 92) ... (t - B,n-1),

where t is a variable, and consequently

m=(1-9)(1-02)...(1The m - 1 factors of the right-hand side are integers and associates of

each other, because if r is one of the numbers 1, 2, ... , m - 1 then

1-or 1

1-e=1+0+0 +...+er-

will be an integer, and if s is positive and chosen so that rs - 1(mod m), then

1-9=9

1-9''e _1 + er + 92r + + e(s-1)r

1 T 1 - or

will also be an integer. Thus if we make the abbreviation

we getm = Eµ.,-1

where a is a unit of the field Il and hence, by taking norms,

m" = [N(,)]m-1

However, since m is a prime number, N(p) must be a power of m. Ifwe put N(p) = me it follows that n = e(m - 1), and since n < m - 1by the remark above, we conclude that e = 1 and n = m - 1 = 0(m).The preceding equation f (9) = 0 is therefore irreducible, the numbers0,0 ..... , 9m-1 are conjugate, and these numbers correspond to m - 1isomorphisms transforming the normal field 1 into itself. At the sametime we have

N(p) = m, om = op".


The principal ideal op is a prime ideal. In fact, if we have op = ab, wherea and b are ideals different from o, it follows that m = N(a)N(b), andsince m is a prime number we necessarily have, say, N(a) = m, N(b) = 1,whence b = o, contrary to hypothesis. At the same time (§21,3) m is thesmallest rational number divisible by p; the numbers 0, 1, 2, ... , m - 1form a complete system of mutually incongruent numbers modulo A. Itfollows from this that a number of the form

w = ko + kip + k2µ2 + ... + km-2µm-2,

where ko, k1i k2, ... , km-2 are integers, is not divisible by m, and hencealso not by µ7a-1, unless all the numbers ko, kl,... , km-2 are divisibleby m. Because an w divisible by m must also be divisible by µ, andthis makes ko divisible by µ, and hence also by µ2, and this makes k1divisible by p, and hence also by m. Continuing in this way, we concludethat the other numbers k1, k2, ... , km-2 are divisible by m.

With the help of this result it is easy to show that the m - 1 numbers1, 9, 92, ... , 9m-2 form a basis for the domain o of all integers in the fieldQ. Since we have

tm - 1 = (t - 1)f (t), mom_1 = (9 - 1)f'(9),

it follows, by excluding the uninteresting case m = 2, that

N[f'(9)] = mm-2,

and since N(9) = 1 and N(9 - 1) = m it follows from §17 that

0(l, 9, 02'. .., 9m-2) _ (-1) 21 mm-2_

Moreover, since p = 1 - 9, 0 = 1 - p it is clear that the two modules[1, 0.... , 9m-2] and [1, µ, ... , µm-2] are identical, whence it follows (§4,3and §17,(5)) that we also have

m-.2'..., 7a-2) _ (-1) 1mm_2.

Since the numbers 1, p, p2, ... , µm-2 are independent, each number inthe field Q can be put in the form

ko + kip + k2p2 + ....+ km-2µ"Z-2 w

k k

where k, ko, k1, k2, ... , km-2 denote rational integers without commondivisor. For this number to be an integer, that is, for w to be divisibleby k, it is necessary (§18) that k2 divide the discriminant of the basis1, µ, µ2, ... , µ7a-2, and therefore k cannot contain prime factors other


than the number m. Moreover, since it has been shown that w is notdivisible by m unless the numbers ko, k1,.. . , k ,,n-2 are divisible by m,k must also not be divisible by m. Thus we must have k = ±1, and allintegers of the field are of the form

w = ko + k1µ + k2µ2 + ... + km-2µm-2,

whence we have

o = [1, L, ... , µm_2] = [1, 0, ... , em-2],

or again, since 1 + 0 + 02 + ... + 9m,-2 + Om_ 1 = 0,

'Mm-2.0 0(Q) = (_1)--r-

Now let p be any prime ideal different from oµ. The positive rationalprime p divisible by p is different from m, and we have

N(p) =pf,where f is the degree of the prime ideal p. Two powers 0', 09 arecongruent modulo such a prime ideal only if they are equal, that is,if r s (mod m). This is because r 0 s (mod m) implies 0' - 03 =or (1- 9''-S) = ep where e is a unit, in which case or cannot be congruentto 93 (mod p). Now since we have (§26,3)

9N(p) = 0 (mod p),

it follows that

pf 1 (mod m).

Let a be the divisor of O(m) = m - 1 to which the number p belongsmodulo m, that is, let a be the least positive exponent for which

pa = 1 (mod m).

As we know, f must be divisible by a, and hence f > a. But since allintegers in the field Il have the form

w = F(9) = x19 + X202 + + x7,,,_19"°-1

where x1i x2, ... , x,,,, are rational integers, it follows from well-knowntheorems, true for every prime number p, that

wP - F(OP), wP' - F(9P'") (mod p),

and henceawP

- w (mod p).


We conclude first of all that the ideal op is a product of distinct primeideals, because if op = p2q there would be a number w divisible by pq butnot by p, so w2 and hence also

wpawould be divisible by p2g2 = pq, and

then also by p, contrary to the preceding congruence. Moreover, since pis divisible by p, every integer w in the field Sl satisfies the congruence

wra - w (mod p),

which therefore has N(p) = pf mutually incongruent roots w. And sinceits degree is pa we must have pf < pa, thus implying f < a. But it hasalready been shown that f > a, so f = a. We have therefore arrived atthe following result, which is the main theorem of Kummer's theory:t

"If p is a prime number different from m and if f is the exponent towhich p belongs modulo m, so that O(m) = e f for some e, then

op = P1p2...pe,

where pl, p2, ... , pe are distinct prime ideals of degree f ."

The rest follows easily. The general case where m is an arbitrarycomposite number can be treated similarly. The degree of the normalfield Sl is always equal to the number O(m) of those numbers among1, 2,3,..., m that are prime to m. The preceding law is proved withoutany change, and the determination of the prime ideals that divide mdoes not present any extra difficulty.

From the very general researches that I am going to publish shortly,the ideals of a normal field Sl immediately allow us to find the idealsof an arbitrary subfield $ of fl, that is, any field H whose members allbelong to Q. For example, this enables us to know the ideals of any fieldH resulting from division of the circle and, to give a more precise ideaof the scope of these researches, I mention the following case.

Again let m be a prime number, so ¢(m) = m - 1, and let e be anydivisor of m -1 = e f . In the theory of rational numbers, the congruence

kf - 1 (mod m)

has precisely f mutually incongruent roots h, which are closed undermultiplication and which, in that sense, form a group. If 0 is again aprimitive root of the equation 0 = 1 and if Sl is the corresponding fieldof degree m - 1, then all the numbers F(9) in this field satisfying the

t Kummer's researches may be found in Crelle's Journal, 35, in Liouville's Journal,XVI and in the memoirs of the Berlin Academy for the year 1856.

t Dedekind calls it a "divisor", as in his footnote in §15. (Translator's note.)


conditions F(9) = F(9h) form a field H of degree e, and the e conjugateperiods§ 771, 772, ... , 17e are sums of f terms, one of which is

9h

and they form a basis of the domain a of integers in H. By generalresults I shall speak of later (or else immediately, by conclusions likethose derived above in the case e = m - 1) we now obtain the primeideals belonging to this subfield H of the normal field Q. If we put

p = H(1 - 9h),

then p is an integer of the field H, m is an associate of pe, and ep is aprime ideal. Moreover, if p is a rational prime different from m, and ifpf belongs to the exponent f modulo m, then f will necessarily be adivisor of e = e' f' and the principal ideal ep will be the product of e'distinct prime ideals of degree f'. In the case e = m - 1, f = 1, H isidentical with Q, and we again obtain the result proved above. We nowexamine more closely the case e = 2, f = 2

In this case the f numbers h are the quadratic residues of m. If welet k denote a quadratic nonresidue, then the two conjugate periods

77=1: 9h 7 =1: 9k

form a basis of the domain e of all integers in the quadratic field H, andhence its discriminant will be

21/ 11

7 /(77

-771)2,0(H)7

because 77 + 77' = -1. The number m is an associate of the square ofthe number p = 11(1 - 9k), and ep is a prime ideal. Moreover, ep is theproduct of two prime ideals of degree 1, or else ep is a prime ideal ofdegree 2, according as

p t = +1 or - 1 (mod m),m 1

that is, using the notation of Legendre, according as

(m) = +1 or - 1.

But we can directly study all quadratic fields, without recourse to divi-sion of the circle, and we have already (§18) determined the discriminantD' of such a field H. From D' we can also derive the prime idealst of

§ Disquisitiones Arithmeticae, art. 343.f See Dirichlet's Vorlesungen fiber Zahlentheorie, § 168.


the field H. If the rational prime p divides D' then the correspondingprincipal ideal ep will be the square of a prime ideal. If p does not divideD', and if p is odd, then ep will be the product of two prime ideals ofdegree 1, or else a prime ideal of degree 2, according as

D' =+1or -1.p

Finally, if D' is odd and hence - 1 (mod 4) then e(2) will be the productof two prime ideals of degree 1, or else a prime ideal of degree 2, accordingas

D' - 1 or 5 (mod 8).

Comparing these laws, valid for all quadratic fields, with the result forthe special field H derived using circle division, we see first that D' mustbe divisible by m, but not by any other prime number, and hence that(§18)

0(H) = D' = (-1)'..2m.

In this way we derive, from entirely general principles and without cal-culation, the result

(77 -77')2m,previously demonstrated in the theory of circle division by effective for-mation of the square of i - i'.t Pursuing this comparison further, weare led again to the theorem

m) = (:,:p )'where ±m = 1 (mod 4), and to the theorem

_ s.Cm)

ma-1(-1)

This proof of the quadratic reciprocity law, in which we also determinethe quadratic character of the number -1, is essentially the same as thecelebrated sixth proof of Gauss, t later reproduced in different forms byJacobi, Eisenstein and others. I should say that it was by meditatingon the essence of that proof and the analogous proofs of cubic and bi-quadratic reciprocity that I was led to the general researches mentionedabove and soon to be published.

t Disquisitiones Arithmeticae, art. 356.t Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et

ampliationes nov&, 1817.


As a final example, we consider the case m = 4. We then have 0 = i =and the integers of the quadratic field 1 are the complex integers,

first introduced by Gauss, of the form

w=x+yi,

where x, y are rational integers (§6). The discriminant of the field is

1 i

1 -i= -4

2

The number 2 = i(i - 1)2 is an associate of the square of the primenumber 1 - i. If p is a positive odd rational prime then we have

iP = (-1) P21 i,

and consequently

WP = (x + yi)P - x + (-1) 2 yi (mod p).

Now if p - 1 (mod 4), each integer w will satisfy the congruence

wP - w (mod p),

whence it follows immediately that op is the product of two differentprime ideals of degree 1. But if p - 3 (mod 4) we have

2wP=w WP -w (modp),

where w' is the number conjugate to w, and we conclude easily that opis a prime ideal of degree 2. But every ideal a of this field must be aprincipal ideal. In fact, if ao is a member of the ideal a with minimumnorm, then each number a in the ideal will be divisible by ao. This isbecause (§6) we can choose the integer w so that

N(a - wao) < N(ao),

and since the numbers a, ao, and hence also a-wao, belong to the ideala, we must have N(a - wao) = 0, whence a = wao and consequentlya = oao. Q.E.D.

Now, since op is the product of two prime ideals of degree 1 in thecase where p is a rational prime =- 1 (mod 4), it follows that

p = N(ao) = N(a + bi) = a2 + b2,

which is the celebrated theorem of Fermat.


§28. Classes of ideals

We now return to consideration of an arbitrary field S2 of degree n,in order to establish the distribution of its ideals into classes. Thisdistribution depends first of all on the theorem (§25,5) that each ideala can be converted into a principal ideal by multiplication by an idealm, and on the following definition: two ideals a, a' are called equivalentwhen they can be converted into principal ideals am = op, am = oµ' bymultiplication by the same ideal m. Then we evidently have µ'a = µa'and, conversely, if there are two nonzero numbers q, if such that 77'a =r7a' then the ideals a, a' are certainly equivalent. Because, multiplying aby m to obtain a principal ideal am = oµ, it follows that oµ77' = i7'am =77a'm. Then p77' is divisible by 77, whence pi7' = µ'r7, op'r7 = r7a'm, hencea'm = oµ'. Q.E.D.

If two ideals a', a" are equivalent to a third, a, then a' and a" are equiv-alent to each other. This is because the hypothesis yields four numbersµ, µ', 77,77 " such that µ'a = pa', 77"a = rra", and hence (r7" µ)a' = (p'77)a".Q.E.D.

It follows that the ideals can be partitioned into classes. If a is a givenideal, the system A of all ideals a, a', a",... equivalent to a will be calleda class of ideals, and a will be called a representative of the class A. Anytwo ideals in A are equivalent, and any ideal a' in A can be chosen as arepresentative in place of a.

It is clear that the system of all principal ideals forms a class by itself,since each of them is converted to itself when multiplied by the idealo, and hence they are all equivalent. And if an ideal a is equivalentto a principal ideal, and hence equivalent to o, then a itself must be aprincipal ideal, since there are two numbers p, µ' such that µ'a = op andthis implies that p is divisible by p', whence p = µ'µ" and consequentlya = op". Thus the class represented by o includes all principal idealsand no others. We call this class the principal class and denote it by 0.

Now if a runs through all the ideals in a class A, and b runs throughall the ideals in a class B, then all the products ab belong to the sameclass K. Because if a', a" are in A and b', b" are in B there are fournumbers a', a", /3', /3" such that a"a' = a'a", /3"b' = Q'b", and it followsthat (a",Q")(a'b') = (a'/')(a"b"), that is, a'b' and a"b" are equivalentideals. We denote the class K to which all the products ab belong byAB and call it the product of A and B, or the class composed from Aand B. We evidently have AB = BA, and it follows from the equation

§29. The number of classes of ideals 147

(ab)c = a(bc) that (AB)C = A(BC) for any three classes A, B, C. Thenwe can apply the same reasoning as for the multiplication of numbersor ideals, and show that, in the composition of any number of classesAl, A2, ... , Am, the order in which pairs of classes are combined has noinfluence on the final result, which we can denote simply by A, A2 A,,,.

If the ideals al, a2i ... , am represent the classes A1, A2, ... , Am then theideal a, a2 an, represents the class Al A2 Am. If all the m factorsequal A, then their product is called the mth power of A, and we denoteit by A'. In addition, we put Al = A and A° = O. The following twocases are particularly important:

The equation oa = a yields the theorem that OA = A for any class A.Moreover, since each ideal a can be transformed into a principal ideal

am by multiplication by an ideal m, for each class A there is a class Msatisfying the condition AM = 0, and only one, because if the class Nis such that AN = 0 it follows that

N = NO = N(AM) = M(AN) = MO = M.

The class M is called the class opposite or inverse to A, and we denoteit by A. Conversely, it is clear that A will be the class inverse to A.If in addition we define A-' to be the class inverse to A' then we havethe following theorems for any rational integer exponents r, s:

ArAs = Ar+s, (Ar)s = Ars, (AB)r = ATBr.

Finally, it is evident that from AB = AC we can always deduce B = C,multiplying by A-'.

§29. The number of classes of idealsIf we take any n integers W1, w2, ... , wn forming a basis for the field Il,then each number

w=hlwl+h2w2+--+hnwn

with rational integer coordinates hl, h2i ... , hn will be an integer in thesame field. If we allow the coordinates to take all integer values ofabsolute value not greater than a particular positive value k, then it isevident that the absolute values of the corresponding numbers w, whichare real, or their analytic moduli, which are imaginary, are all < rkwhere r is the sum of the absolute values or moduli of w1, w2, ... , wn,


and hence a constant independent of k. Moreover, since the norm N(w)is a product of n conjugate numbers w of the form above, we also have

±N(w) < skn,

where s is likewise a constant depending only on the basis. We deducefrom this the following theorem:

In each class M of ideals there is at least one ideal m whose norm isbounded by a constant.

Proof. Take any ideal a in the inverse class M-1, and let k be thepositive rational integer determined by the conditions

kn < N(a) < (k + 1)n.

If we now allow each of the n coordinates hl, h2i ... , hn to take all k + 1values 0, 1, 2, ... , k, then we obtain distinct numbers w and, since theirnumber is (k+1)n and hence > N(a), there are necessarily two differentnumbers w,

a = blwl + ... + brawn 'Y = C1w1 + ... + Cnwn,

which are congruent modulo a. Hence their difference

a = (bl - Cl)wl + ... + (bra - Cn)wn

will be a nonzero number divisible by a. But since the coordinates b, c ofthe numbers ,Q, 'y come from the sequence 0, 1, 2, ... , k, the coordinatesb - c of a all have absolute value not greater than k, and hence

±N(a) < skn.

Since a is divisible by a we have oa = am, where m is an ideal in theclass M, and hence

±N(a) = N(a)N(m) < skn.

Moreover, since kn < N(a), it follows that N(m) < s. Q.E.D.

If we now consider that the norm m of an ideal m is always divisibleby m (§20), it is clear that there cannot be more than a finite numberof ideals m with given norm m, because each ideal, and in particularom, is divisible by only a finite number of ideals (§25,8). Since there arealso only a finite number of rational integers m not exceeding a givenconstant s, there cannot be more than a finite number of ideals m withN(m) < s, which evidently yields the fundamental theorem:


The number of classes of ideals of the field St is finite.

The exact determination of the number of classes of ideals is incon-testably a very important problem, but also one of the most difficultin the theory of numbers. For quadratic fields, whose theory essentiallycoincides with that of binary quadratic forms, we know that the problemwas first solved by Dirichlet.t His solution, expressed in the terminologyof the theory of ideals, rests on the study of the function

1 _7T 1

N(a)s -11 1- N(P),

for infinitely small positive values of the independent variable s - 1.The sum is taken over all ideals a, the product over all prime ideals p,and the identity of the two expressions is an immediate consequence ofthe laws of divisibility. With the aid of these principles, the number ofclasses of forms or ideals has been later determined by Eisensteint fora particular case of a field of degree 3, and by Kummer$ for the higherdegree fields arising from division of the circle. These researches haveexcited the liveliest interest because of their astonishing connections withanalysis, algebra and other parts of number theory. For example, theproblem treated by Kummer is closely related to Dirichlet's proof of thetheorem on primes in arithmetic progressions, which can be considerablysimplified with the aid of these researches. There is no doubt that furtherstudy of the general problem will lead to important progress in thesebranches of mathematics; however, while part of this research has beensuccessfully completed for an arbitrary field St,§ we are nevertheless farfrom the complete solution, and for the moment we are confined tostudying new special cases.

§30. ConclusionWe shall derive some further interesting consequences of the fundamentaltheorem proved above. (See Disquisitiones Arithmeticae, art. 305-307.)

Let h be the number of classes of ideals of the field S2, and let A be aparticular class. The h + 1 powers

0, A,A2,...,Ah-i,A'

t Crelle's Journal, 19, 21.t Crelle's Journal, 28.t Crelle's Journal, 40, Liouville's Journal, XVI.§ Dirichlet, Vorlesungen fiber Zahlentheorie, §167.


cannot all be different, hence there are two different exponents r andr + m > r in the sequence 0, 1, 2, ... , h such that Ar+m = Ar, andconsequently

Am=0.Moreover, if m is the smallest positive exponent satisfying this conditionit is easy to see that the m classes

0, A,A2,...,A--1

are all different, and we say that the class A belongs to the exponent m.Obviously Al-1 = A-1, and more generally we have Ar = A8 if andonly if r - s (mod m). Then, if B denotes any class, the m classes

(B) B, BA, BA 2'. .., BA--1

will be all different, and any two complexes of m classes, such as thepreceding (B) and the following

(C) C, CA, CA2,... , CAm-1,

will either be identical or have nothing in common. In fact, if bothinclude the same class BAr = CA8 then we have C = BAr-3, whenceit follows immediately that the m classes in (C) are the same as thosein (B). Thus the whole system of h classes is partitioned into g suchcomplexes and, since each complex includes m different classes, we haveh = mg. That is, the exponent m to which the class A belongs is alwaysa divisor of the number of classes, h. It follows that we have the theorem

Ah=O

for each class A. Now if a is any ideal in any class A, then ah belongsto the class Ah, and hence to the principal class. That is, the ht' powerof any ideal is a principal ideal.

With this important theorem we come to see the notion of ideal froma new point of view, at the same time connected with a precise definitionof ideal numbers. Let a be any ideal and let ah = oat. Now if a denotesany number in the ideal a, ah will be in ah, and hence divisible by thenumber al, and it follows from §13,2 that a is divisible by the integerp = a1, which does not in general belong to the field Q. Conversely, ifa is an integer belonging to S2 and divisible by p, then ah will be divisibleby µh = a1 and consequently (oa)h will be divisible by oat = ah, fromwhich we easily conclude, by the general laws of divisibility (§25), thatoa is divisible by a, so that a is a number in the ideal a. Thus the ideal


a consists of all the integers in SZ divisible by the integer p. For thisreason we say that the number p, though not actually a member of Q,is an ideal number of the field ci, and that it corresponds to the ideal a.Or, a little more generally, an algebraic integer u is said to be an idealnumber of the field SZ when there is a power p', with positive integerexponent r, equal to an actual rl in St, and at the same time an ideal ain the field St satisfies the condition aT = op. The latter ideal a is theideal corresponding to the ideal number p, and it is a principal ideal ifand only if p is an associate of an actual number in the field Q. (See theIntroduction and §10.)

We end our considerations with the proof of the following theoremannounced earlier (§14):

Any two algebraic integers a, 0 have a common divisor 6 which canbe expressed in the form 6 = aa' + 30', where a' and ,3' are likewisealgebraic integers.

Proof. We assume that the two numbers a, ,Q are nonzero, otherwisethe theorem is evident. Then it is easy to see that there is a field ci offinite degree including both a, 3. Let o be the domain of integers of thisfield, and let h be the number of classes of ideals. Now put

oa = act, 00 = b,0, ah = 0151,

where a is the greatest common divisor of oa, o,3, and 61 is in o. Sinceah, ,13h are divisible by oh, we can put

ah = a161, Nh = )3161, Dal = ah, 0,31 = bh,

where al, ,31 are likewise in o. Also, since a and b are relatively primeideals, o will be the greatest common divisor of oal, 0/31 and, since thenumber 1 is in o, there will be two numbers a2, 32 in o satisfying thecondition

ala2 + /31/32 = 1, or aha2 +,3h,32 = 61.

If we now put

61 = 6h,

then the integer 6 will be a common divisor of a and ,3, since ah, ,(3h aredivisible by 61i and hence, since h > 1, we can put

a2ah-1 = a'6h-1, ,02/3h-1 =,Ql6h-1,

where a', j3' are integers satisfying the condition aa' +,130' = 6. Q.E.D.


If at least one of the two numbers a, /3 is nonzero, then the number6, and any of its associates, deserves the name greatest common divisorof a, 3. If 6 is a unit then a, /3 may be called relatively prime, andtwo such numbers enjoy the characteristic property that any number pdivisible by a and /3 is also divisible by a/3. This is because the equationsµ = aa" = /3/3" and 1 = aa' + /3/3' imply

p = a/3(a'/3" +,3'a"),

and the converse is equally valid, since a, /3 are both nonzero.

Index

abelian group, 20adjoined element, 94algebraic integers, 3, 53, 54, 103

closure properties, 4, 40, 103congruent, 55decomposable, 55definition, 4, 39divisibility, 54, 105greatest common divisor, 106, 152prime, 54relatively prime, 152units, 54, 106

algebraic number, 53, 103conjugate, 55definition, 39norm, 54, 111

Arithmeticaof Diophantus, 8

Artin, 3, 46associates, 85, 106associativity

of composition, 20automorphism

of fields, 35

Baker, 42basis

coordinates with respect to, 108of cyclotomic integers, 140of field, 108of module, 67

belonging to an exponent, 150binomial theorem

mod p, 11, 36biquadratic

reciprocity, 144residues, 84

Brahmaguptaidentity, 18, 26, 28

characterbiquadratic, 38cubic, 38quadratic, 33

circle division, 28, 31, 55, 138class

ideal number, 56modulo a module, 64modulo an algebraic integer, 118modulo an ideal, 97, 122number, 55, 61of Gaussian integers, 84, 86of ideals, 61, 146of rational integers, 84principal, 56

class group, 19, 20class number, 41, 61

determination of, 149Dirichlet formula, 42finiteness, 42, 149of cubic field, 149of cyclotomic field, 42, 149of number field, 41of Q(/), 41of quadratic fields, 149of quadratic forms, 14, 56

commutativityof composition, 20

complete systemof incongruent numbers, 122of representatives, 65

complex integers, 53, 84composite

Gaussian integer, 85rational integer, 83

compositionasociativity of, 20commutativity of, 20Gauss definition, 20Legendre definition, 20

153

154

of forms, 17, 19, 28, 44, 100, 102of ideal classes, 146

congruencehigher order, 57modulo a module, 64modulo an algebraic integer, 118modulo an ideal, 97, 121of algebraic integers, 55of Gaussian integers, 85of quadratic integers, 87of rational integers, 84roots of, 137

conjugate

algebraic numbers, 55and norm, 112fields, 110ideals, 97in quadratic field, 89, 116number, 41, 110numbers in quadratic field, 87periods, 143

constructionstraightedge and compass, 31, 32

continued fraction, 4continuity, 58coordinates

with respect to a basis, 108correspondence

of numbers and ideals, 121cubic

field, 117reciprocity, 144

cyclotomicequation, 31, 32, 139field, 139integers, 30, 32, 140

decomposable numbers, 55Dedekind

and Weber theory, 46avoidance of symmetric functions, 41class number theorem, 42definition of algebraic integer, 39domain, 5invention of ideals, 3proof of quadratic reciprocity, 37proof of two square theorem, 25section, 44, 58Supplement to Dirichlet, 20theory of ideals, 29, 37

degreeof a field, 54, 108of a form, 56of a prime ideal, 124

Descartes, 10determinant, 71

of a quadratic form, 14, 98

Index

Diophantine equation, 8Diophantus, 8

Arithmetica, 8identity, 9, 18, 23

Dirichlet, 56, 149class number formula, 42theorem on primes, 42, 149Vorlesungen, 5, 21, 40, 45, 53, 61, 84,

87, 98, 102, 119, 125, 135-137,143, 149

discriminant, 112determines quadratic field, 117is rational, 113of cyclotomic field, 141of field, 116of Gaussian field, 145of quadratic form, 14

Disquisitionesof Gauss, 7, 33, 40

divisibilityby ideal number, 58by nth power, 90of algebraic integers, 54, 105of ideals, 60, 98, 120of modules, 63of quadratic integers, 87of rational integers, 83

divisor, 7behaviour as a, 89ideal, 60of a module, 63of an ideal, 133

domain, 5Dedekind, 5

Eisenstein, 40, 144, 149elementary transformations, 77Elements

of Euclid, 6, 7, 53equation

cyclotomic, 31, 32, 139Pell, 4, 8

hagorean, 29

XT+ y3 = z3, 30x4 + y4 = z2, 10

x4 + y4 = z4, 10

y3=x2+2,9,21equivalence

of ideal factors, 28of quadratic forms, 13

equivalentideal numbers, 56ideals, 146

Euclid, 53algorithm for gcd, 12, 84Elements, 6, 7, 53formula for Pythagorean triples, 6, 10

Index 155

Euclidean algorithm, 12, 84for quadratic integers, 22in Gaussian integers, 23in Z[V/--2], 26

Euler, 9conjecture on x2 + 5y2, 13, 17conjectured quadratic reciprocity, 16criterion, 17, 37proof of two square theorem, 11

Fermat, 9and x2 + 5y2, 6conjecture on x2 + 5y2, 13, 17last theorem, 10, 29, 38last theorem for n = 3, 30last theorem for n = 4, 29little theorem, 11, 15, 137notes on Diophantus, 10, 29numbers, 31primes, 31two square theorem, 9, 11, 85, 145use of Pythagorean triples, 10

fieldautomorphism, 35basis of, 108closure properties, 5, 109conjugate, 110cubic, 117cyclotomic, 139definition, 107degree of, 108discriminant of, 116fundamental number of, 116Galois, 111normal, 111, 116of degree n, 54of finite degree, 5, 39, 41, 106, 108quadratic, 116, 143

finitely generated module, 67, 95forms

binary quadratic, 56of degree n, 56

fundamental number, 116

Galois, 61, 94, 137, 138field, 111

Galois theory, 35, 45Gauss, 6, 53, 77, 98, 102, 137, 144

complex integers, 84composition of forms, 28definition of composition, 20Disquisitiones, 7, 33, 40, 77, 84, 100,

127, 144, 149existence of primitive roots, 16proofs of quadratic reciprocity, 37proved quadratic reciprocity, 16sums, 32

Gaussian integers, 22, 84, 97, 145class, 86composite, 85congruence, 85Euclidean algorithm, 23laws of divisibility, 85norm, 22, 85prime, 85unique prime factorisation, 22, 24, 85units, 23, 85

Gaussian primes, 23and two square theorem, 24

greatest common divisorEuclidean algorithm, 84of algebraic integers, 106, 152of ideals, 121, 134of modules, 63of rational integers, 84

group, 82, 142abelian, 20class, 19, 20

Hilbert Zahlbericht, 38, 46

ideal, 5, 57, 58class of, 61, 146class representative, 146classes modulo, 122congruence modulo, 97, 121conjugate, 97defining properties, 96divisors of, 60, 133fundamental properties, 59in field of degree n, 119norm of, 97, 122numbers, 57power of, 125, 150prime, 60, 101, 123prime factors, 56principal, 59, 97, 120

ideal numberclass, 56definition, 150equivalent, 56of quadratic field, 90

idealsdivisibility of, 98, 120equivalent, 146greatest common divisor, 121, 134in Z, 7, 120least common multiple, 121, 134multiplication of, 60, 98, 125of quadratic integers, 95product of, 43, 44, 60, 125relatively prime, 126, 134unique prime factorisation, 102, 130

identity

156

Brahmagupta, 18, 26, 28Diophantus, 9, 18, 23isomorphism, 110Lagrange, 18

independent numbers, 71, 108infinite descent, 11infinity

horror of, 44integers

algebraic, 54, 103complex, 53cubic, 35cyclotomic, 30, 32, 140Gaussian, 22, 84, 97imaginary quadratic, 25in field of degree n, 113Kummer, 40, 43of Q(a), 5of quadratic field, 102, 116quadratic, 21rational, 4, 83

inverseideal class, 147isomorphism, 110mod p, 12, 34substitution, 13

irrational numbers, 8, 57irreducible

equation, 54, 107polynomial, 107system, 107

irreducible system, 71isomorphism, 108

determined by choice of root, 110fixes rational numbers, 110identity, 110inverse, 110

onto number field, 41

Jacobi, 144

Kronecker, 94abelian group axioms, 21adjoined ideal numbers, 94opposition to infinity, 44theory of fields, 46

Kummer, 6, 55, 83, 86, 90, 94, 138, 149ideal factors, 28integers, 40, 43main theorem, 142

Lagrange, 6identity, 18proof of two squares theorem, 15reduction process, 41

laws of divisibilityin Gaussian integers, 85

Index

in quadratic integers, 86in rational integers, 85, 88

least common multipleof ideals, 121, 134of modules, 63

Legendredefinition of composition, 20notation, 143symbol, 33

linear algebra, 41Lipschitz, 44

module, 62as a group, 82basis of, 67closure properties, 5

congruence modulo, 64

divisor of, 63finitely generated, 67, 95multiple of, 63zero, 63

modulesdivisibility of, 63greatest common divisor of, 63least common multiple of, 63multiplication of, 98

modulus, 55, 84, 118multiple, 7

of algebraic integer, 105of module, 63

multiplicationof ideals, 60, 98, 125of modules, 98

multiplicative propertyof quadratic character, 34

Newton, 40Noether, 3, 46norm

and conjugates, 41, 112in field of finite degree, 41in Z[V ], 27in Zf / , 25is rational, 113multiplicative property, 23, 26, 41,

132of algebraic number, 54, 111of Gaussian integer, 22, 85of ideal, 97, 122of principal ideal, 122of quadratic integer, 87

normal field, 111, 116number

algebraic, 39, 53, 103classes, 55conjugate, 110ideal, 57

Index

irrational, 8, 57

order, 127of a group element, 16of quadratic forms, 127

Pell equation, 4, 8periods, 143

conjugate, 143permutation, 108Plimpton 322, 6prime, 7

divisor property, 7, 22, 23Fermat, 31Gaussian, 23, 85ideal, 101in algebraic integers, 54

in o 5

in 2[(-'2 , 26rational, 83

prime factorsideal, 56

prime ideal, 60, 101, 123degree of, 124has principal ideal multiple, 130of quadratic field, 144

primitive root, 16, 33, 137, 139, 142principal

class, 56

ideal, 59, 97, 120ideal class, 146

productof ideal classes, 146of ideals, 60, 125

Pythagoras, 6, 8theorem, 8

Pythagorean triples, 6and right-angled triangles, 8Euclid's formula, 6, 10primitive, 10used by Fermat, 10

quadraticcharacter, 33character of -1, 17, 144character symbol, 33fields, 143reciprocity, 16, 36, 37, 144residues, 16, 143

quadratic field, 116class number, 149conjugate in, 89, 116discriminant of, 117failure of unique prime factorisation,

87ideal number of, 90integers of, 102, 116

prime ideals of, 144units, 86

quadratic forms, 102x2 + 2xy + 3y2, 15,17,28x2 + 2y2, 12, 15, 26x2 + 3y2, 12, 15x2 + 5yy2, 6, 12, 15, 17, 27x2+y , 12, 15,25class number, 14composition of, 17, 19, 100, 102determinant of, 14, 98discriminant of, 14equivalent, 13inequivalent, 14, 15, 27orders of, 127reduced, 14reduction, 41

quadratic integers, 21and cyclotomic integers, 32congruence, 87divisibility, 87Euclidean algorithm, 22ideals, 95laws of divisibility, 86norm, 87

rationalcomposite number, 83integers, 4, 53, 83numbers, 107operations, 107prime, 83units, 83

rational numbersfield of, 107fixed by isomorphism, 110

reciprocitybiquadratic, 144cubic, 38, 144laws, 28, 38quadratic, 36, 37, 144

reduced quadratic forms, 14relatively prime

algebraic integers, 152ideals, 126, 134

representativeof ideal class, 146of number class, 65

representativescomplete system, 65

residuesbiquadratic, 84quadratic, 143

Riemann, 45surfaces, 46

ring, 4closure properties, 5

157

158 Index

rootof congruence, 137of unity, 28, 38primitive, 33, 137, 139, 142

Schonemann, 137section, 44, 58Serret, 137Stark, 42subfield, 142substitution, 108

inverse, 13unimodular, 14

symmetric functions, 41, 113Newton theorem, 40

two square theorem, 9and Gaussian primes, 24Dedekind proof, 25, 145Euler proof, 11Fermat proof, 11Lagrange proof , 15

unimodular substitution, 14unique prime factorisation

and equivalence of forms, 28failure in [v/-], 30failure in Z[./- ], 5, 27failure in Z[C23], 32failure in cyclotomic integers, 56failure in quadratic field, 87for ideals, 3, 130in complex integers, 56in Disquisitiones, 7in Gaussian integers, 22, 24, 85in rational integers, 22, 56, 84in Z[v/- ], 26in Z[C3], 30of ideals, 5, 102

unitsin algebraic integers, 54, 106in Gaussian integers, 23, 85in quadratic field, 86in rational integers, 83

vector space, 41

Weber, 45and Dedekind, 46

Weil, 13

zero module, 63

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Richard Dedekind Theory of Algebraic Integers