Splitting Field

Splitting eldFrom Wikipedia, the free encyclopedia

Contents

1 Abelian extension 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Algebraic closure 22.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Existence of an algebraic closure and splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Separable closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Algebraic extension 43.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Degree of a eld extension 64.1 Denition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 The multiplicativity formula for degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.2.1 Proof of the multiplicativity formula in the nite case . . . . . . . . . . . . . . . . . . . . 74.2.2 Proof of the formula in the innite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Dual basis in a eld extension 9

6 Field extension 106.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.5 Algebraic and transcendental elements and extensions . . . . . . . . . . . . . . . . . . . . . . . . 11

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ii CONTENTS

6.6 Normal, separable and Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.8 Extension of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Finite eld 147.1 Denitions, rst examples, and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Explicit construction of nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.3.1 Non-prime elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3.2 Field with four elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3.3 GF(p2) for an odd prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3.4 GF(8) and GF(27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.3.5 GF(16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7.4 Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4.1 Discrete logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.5 Frobenius automorphism and Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.6 Polynomial factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.6.1 Irreducible polynomials of a given degree . . . . . . . . . . . . . . . . . . . . . . . . . . 207.6.2 Number of monic irreducible polynomials of a given degree over a nite eld . . . . . . . . 21

7.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.8.1 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.8.2 Wedderburns little theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Galois extension 248.1 Characterization of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9 Normal extension 269.1 Equivalent properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS iii

9.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.3 Normal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10 Ring homomorphism 2810.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.3 The category of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

10.3.1 Endomorphisms, isomorphisms, and automorphisms . . . . . . . . . . . . . . . . . . . . . 3010.3.2 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

11 Separable extension 3111.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.5 The denition of separable non-algebraic extension elds . . . . . . . . . . . . . . . . . . . . . . 3311.6 Dierential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

12 Simple extension 3612.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.2 Structure of simple extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13 Splitting eld 3813.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.2 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3 Constructing splitting elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.2 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.3 The eld Ki[X]/(f(X)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

13.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.1 The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4.2 Cubic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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13.4.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

14 Tower of elds 4214.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 43

14.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter 1

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois groupis a cyclic group, we have a cyclic extension. A Galois extension is called solvable if its Galois group is solvable, i.e.if it is constructed from a series of abelian groups corresponding to intermediate extensions.Every nite extension of a nite eld is a cyclic extension. The development of class eld theory has provided detailedinformation about abelian extensions of number elds, function elds of algebraic curves over nite elds, and localelds.There are two slightly dierent concepts of cyclotomic extensions: these can mean either extensions formed byadjoining roots of unity, or subextensions of such extensions. The cyclotomic elds are examples. Any cyclotomicextension (for either denition) is abelian.If a eld K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resultingso-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n,since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th rootsof elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-directproduct. The Kummer theory gives a complete description of the abelian extension case, and the KroneckerWebertheorem tells us that if K is the eld of rational numbers, an extension is abelian if and only if it is a subeld of a eldobtained by adjoining a root of unity.There is an important analogy with the fundamental group in topology, which classies all covering spaces of a space:abelian covers are classied by its abelianisation which relates directly to the rst homology group.

1.1 References Kuz'min, L.V. (2001), cyclotomic extension, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

1

Chapter 2

Algebraic closure

For other uses, see Closure (disambiguation).

In mathematics, particularly abstract algebra, an algebraic closure of a eld K is an algebraic extension of K that isalgebraically closed. It is one of many closures in mathematics.Using Zorns lemma, it can be shown that every eld has an algebraic closure,[1][2][3] and that the algebraic closureof a eld K is unique up to an isomorphism that xes every member of K. Because of this essential uniqueness, weoften speak of the algebraic closure of K, rather than an algebraic closure of K.The algebraic closure of a eld K can be thought of as the largest algebraic extension of K. To see this, note that if Lis any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is containedwithin the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed eld containingK, because ifM is any algebraically closed eld containing K, then the elements ofM that are algebraic over K forman algebraic closure of K.The algebraic closure of a eld K has the same cardinality as K if K is innite, and is countably innite if K is nite.[3]

2.1 Examples The fundamental theorem of algebra states that the algebraic closure of the eld of real numbers is the eld ofcomplex numbers.

The algebraic closure of the eld of rational numbers is the eld of algebraic numbers.

There are many countable algebraically closed elds within the complex numbers, and strictly containing theeld of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers,e.g. the algebraic closure of Q().

For a nite eld of prime power order q, the algebraic closure is a countably innite eld that contains a copyof the eld of order qn for each positive integer n (and is in fact the union of these copies).[4]

2.2 Existence of an algebraic closure and splitting eldsLet S = ffj 2 g be the set of all monic irreducible polynomials in K[x]. For each f 2 S , introduce newvariables u;1; : : : ; u;d where d = degree(f) . Let R be the polynomial ring over K generated by u;i for all 2 and all i degree(f) . Write

f dY

i=1

(x u;i) =d1Xj=0

r;j xj 2 R[x]

2

2.3. SEPARABLE CLOSURE 3

with r;j 2 R . Let I be the ideal in R generated by the r;j . By Zorns lemma, there exists a maximal idealM in Rthat contains I. Now R/M is an algebraic closure of K; every f splits as the product of the x (u;i +M) .The same proof also shows that for any subset S of K[x], there exists a splitting eld of S over K.

2.3 Separable closureAn algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separableextensions ofK withinKalg. This subextension is called a separable closure ofK. Since a separable extension of a sep-arable extension is again separable, there are no nite separable extensions of Ksep, of degree > 1. Saying this anotherway, K is contained in a separably-closed algebraic extension eld. It is essentially unique (up to isomorphism).[5]

The separable closure is the full algebraic closure if and only if K is a perfect eld. For example, if K is a eld ofcharacteristic p and if X is transcendental over K,K(X)( p

pX) K(X) is a non-separable algebraic eld extension.

In general, the absolute Galois group of K is the Galois group of Ksep over K.[6]

2.4 See also Algebraically closed eld Algebraic extension Puiseux expansion

2.5 References[1] McCarthy (1991) p.21

[2] M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp.11-12.

[3] Kaplansky (1972) pp.74-76

[4] Brawley, Joel V.; Schnibben, George E. (1989), 2.2 The Algebraic Closure of a Finite Field, Innite Algebraic Extensionsof Finite Fields, Contemporary Mathematics 95, American Mathematical Society, pp. 2223, ISBN 978-0-8218-5428-0,Zbl 0674.12009.

[5] McCarthy (1991) p.22

[6] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.

Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. ISBN 0-226-42451-0. Zbl 1001.16500.

McCarthy, Paul J. (1991). Algebraic extensions of elds (Corrected reprint of the 2nd ed.). New York: DoverPublications. Zbl 0768.12001.

Chapter 3

Algebraic extension

In abstract algebra, a eld extension L/K is called algebraic if every element of L is algebraic over K, i.e. if everyelement of L is a root of some non-zero polynomial with coecients in K. Field extensions that are not algebraic, i.e.which contain transcendental elements, are called transcendental.For example, the eld extensionR/Q, that is the eld of real numbers as an extension of the eld of rational numbers,is transcendental, while the eld extensionsC/R andQ(2)/Q are algebraic, whereC is the eld of complex numbers.All transcendental extensions are of innite degree. This in turn implies that all nite extensions are algebraic.[1] Theconverse is not true however: there are innite extensions which are algebraic. For instance, the eld of all algebraicnumbers is an innite algebraic extension of the rational numbers.If a is algebraic over K, then K[a], the set of all polynomials in a with coecients in K, is not only a ring but a eld:an algebraic extension of K which has nite degree over K. The converse is true as well, if K[a] is a eld, then a isalgebraic over K. In the special case where K =Q is the eld of rational numbers, Q[a] is an example of an algebraicnumber eld.A eld with no nontrivial algebraic extensions is called algebraically closed. An example is the eld of complexnumbers. Every eld has an algebraic extension which is algebraically closed (called its algebraic closure), but provingthis in general requires some form of the axiom of choice.An extension L/K is algebraic if and only if every sub K-algebra of L is a eld.

3.1 PropertiesThe class of algebraic extensions forms a distinguished class of eld extensions, that is, the following three propertieshold:[2]

1. If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.

2. If E and F are algebraic extensions of K in a common overeld C, then the compositum EF is an algebraicextension of K.

3. If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K.

These nitary results can be generalized using transnite induction:

1. The union of any chain of algebraic extensions over a base eld is itself an algebraic extension over the samebase eld.

This fact, together with Zorns lemma (applied to an appropriately chosen poset), establishes the existence of algebraicclosures.

4

3.2. GENERALIZATIONS 5

3.2 GeneralizationsMain article: Substructure

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding ofM into N is calledan algebraic extension if for every x in N there is a formula p with parameters inM, such that p(x) is true and the set

ny 2 N

p(y)ois nite. It turns out that applying this denition to the theory of elds gives the usual denition of algebraic extension.The Galois group of N overM can again be dened as the group of automorphisms, and it turns out that most of thetheory of Galois groups can be developed for the general case.

3.3 See also Integral element Lroths theorem Galois extension Separable extension Normal extension

3.4 Notes[1] See also Hazewinkel et al. (2004), p. 3.

[2] Lang (2002) p.228

3.5 References Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhalovna; Kirichenko, Vladimir V. (2004),Algebras, rings and modules 1, Springer, ISBN 1-4020-2690-0

Lang, Serge (1993), V.1:Algebraic Extensions, Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub.Co., pp. 223, ISBN 978-0-201-55540-0, Zbl 0848.13001

McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of elds, New York:Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001

Roman, Steven (1995), Field Theory, GTM 158, Springer-Verlag, ISBN 9780387944081 Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 9780130878687

Chapter 4

Degree of a eld extension

In mathematics, more specically eld theory, the degree of a eld extension is a rough measure of the size ofthe eld extension. The concept plays an important role in many parts of mathematics, including algebra and numbertheory indeed in any area where elds appear prominently.

4.1 Denition and notation

Suppose that E/F is a eld extension. Then E may be considered as a vector space over F (the eld of scalars). Thedimension of this vector space is called the degree of the eld extension, and it is denoted by [E:F].The degree may be nite or innite, the eld being called a nite extension or innite extension accordingly. Anextension E/F is also sometimes said to be simply nite if it is a nite extension; this should not be confused with theelds themselves being nite elds (elds with nitely many elements).The degree should not be confused with the transcendence degree of a eld; for example, the eld Q(X) of rationalfunctions has innite degree over Q, but transcendence degree only equal to 1.

4.2 The multiplicativity formula for degrees

Given three elds arranged in a tower, say K a subeld of L which is in turn a subeld ofM, there is a simple relationbetween the degrees of the three extensions L/K, M/L and M/K:

[M : K] = [M : L] [L : K]:

In other words, the degree going from the bottom to the top eld is just the product of the degrees going fromthe bottom to the middle and then from the middle to the top. It is quite analogous to Lagranges theorem ingroup theory, which relates the order of a group to the order and index of a subgroup indeed Galois theory showsthat this analogy is more than just a coincidence.The formula holds for both nite and innite degree extensions. In the innite case, the product is interpreted in thesense of products of cardinal numbers. In particular, this means that if M/K is nite, then both M/L and L/K arenite.IfM/K is nite, then the formula imposes strong restrictions on the kinds of elds that can occur betweenM andK, viasimple arithmetical considerations. For example, if the degree [M:K] is a prime number p, then for any intermediateeld L, one of two things can happen: either [M:L] = p and [L:K] = 1, in which case L is equal to K, or [M:L] = 1 and[L:K] = p, in which case L is equal toM. Therefore there are no intermediate elds (apart fromM and K themselves).

6

4.2. THE MULTIPLICATIVITY FORMULA FOR DEGREES 7

4.2.1 Proof of the multiplicativity formula in the nite caseSuppose that K, L andM form a tower of elds as in the degree formula above, and that both d = [L:K] and e = [M:L]are nite. This means that we may select a basis {u1, ..., ud} for L over K, and a basis {w1, ..., we} forM over L. Wewill show that the elements umwn, for m ranging through 1, 2, ..., d and n ranging through 1, 2, ..., e, form a basis forM/K; since there are precisely de of them, this proves that the dimension of M/K is de, which is the desired result.First we check that they spanM/K. If x is any element ofM, then since the wn form a basis forM over L, we can ndelements an in L such that

x =eX

n=1

anwn = a1w1 + + aewe:

Then, since the um form a basis for L over K, we can nd elements bm,n in K such that for each n,

an =dX

m=1

bm;num = b1;nu1 + + bd;nud:

Then using the distributive law and associativity of multiplication in M we have

x =eX

n=1

dX

m=1

bm;num

!wn =

eXn=1

dXm=1

bm;n(umwn);

which shows that x is a linear combination of the umwn with coecients from K; in other words they spanM over K.Secondly we must check that they are linearly independent over K. So assume that

0 =eX

n=1

dXm=1

bm;n(umwn)

for some coecients bm,n in K. Using distributivity and associativity again, we can group the terms as

0 =eX

n=1

dX

m=1

bm;num

!wn;

and we see that the terms in parentheses must be zero, because they are elements of L, and the wn are linearlyindependent over L. That is,

0 =dX

m=1

bm;num

for each n. Then, since the bm,n coecients are in K, and the um are linearly independent over K, we must have thatbm,n = 0 for all m and all n. This shows that the elements umwn are linearly independent over K. This concludes theproof.

4.2.2 Proof of the formula in the innite caseIn this case, we start with bases u and w of L/K and M/L respectively, where is taken from an indexing set A,and from an indexing set B. Using an entirely similar argument as the one above, we nd that the products uwform a basis for M/K. These are indexed by the cartesian product A B, which by denition has cardinality equal tothe product of the cardinalities of A and B.

8 CHAPTER 4. DEGREE OF A FIELD EXTENSION

4.3 Examples The complex numbers are a eld extension over the real numbers with degree [C:R] = 2, and thus there are nonon-trivial elds between them.

The eld extension Q(2, 3), obtained by adjoining 2 and 3 to the eld Q of rational numbers, has degree4, that is, [Q(2, 3):Q] = 4. The intermediate eld Q(2) has degree 2 over Q; we conclude from themultiplicativity formula that [Q(2, 3):Q(2)] = 4/2 = 2.

The nite eld GF(125) = GF(53) has degree 3 over its subeld GF(5). More generally, if p is a prime and n,m are positive integers with n dividing m, then [GF(pm):GF(pn)] = m/n.

The eld extensionC(T)/C, whereC(T) is the eld of rational functions overC, has innite degree (indeed it isa purely transcendental extension). This can be seen by observing that the elements 1, T, T2, etc., are linearlyindependent over C.

The eld extension C(T2) also has innite degree over C. However, if we view C(T2) as a subeld of C(T),then in fact [C(T):C(T2)] = 2. More generally, if X and Y are algebraic curves over a eld K, and F : X Yis a surjective morphism between them of degree d, then the function elds K(X) and K(Y) are both of innitedegree over K, but the degree [K(X):K(Y)] turns out to be equal to d.

4.4 GeneralizationGiven two division rings E and F with F contained in E and the multiplication and addition of F being the restrictionof the operations in E, we can consider E as a vector space over F in two ways: having the scalars act on the left,giving a dimension [E:F], and having them act on the right, giving a dimension [E:F]. The two dimensions need notagree. Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above appliesto left-acting scalars without change.

4.5 References page 215, Jacobson, N. (1985). Basic Algebra I. W. H. Freeman and Company. ISBN 0-7167-1480-9. Proofof the multiplicativity formula.

page 465, Jacobson, N. (1989). Basic Algebra II. W. H. Freeman and Company. ISBN 0-7167-1933-9. Brieydiscusses the innite dimensional case.

Chapter 5

Dual basis in a eld extension

In mathematics, the linear algebra concept of dual basis can be applied in the context of a nite extension L/K, byusing the eld trace. This requires the property that the eld trace TrL/K provides a non-degenerate quadratic formover K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect eld, and hencein the cases where K is nite, or of characteristic zero.A dual basis is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using asecond basis for computations.Consider two bases for elements in a nite eld, GF(pm):

B1 = 0; 1; : : : ; m1

and

B2 = 0; 1; : : : ; m1

then B2 can be considered a dual basis of B1 provided

Tr(i j) =0; if i 6= j1; otherwise

Here the trace of a value in GF(pm) can be calculated as follows:

Tr() =m1Xi=0

pi

Using a dual basis can provide a way to easily communicate between devices that use dierent bases, rather than havingto explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implementedthen conversion from an element in the original basis to the dual basis can be accomplished with a multiplication bythe multiplicative identity (usually 1).

9

Chapter 6

Field extension

In abstract algebra, eld extensions are the main object of study in eld theory. The general idea is to start with abase eld and construct in some manner a larger eld that contains the base eld and satises additional properties.For instance, the set Q(2) = {a + b2 | a, b Q} is the smallest extension of Q that includes every real solution tothe equation x2 = 2.

6.1 Denitions

Let L be a eld. A subeld of L is a subset K of L that is closed under the eld operations of L and under takinginverses in L. In other words, K is a eld with respect to the eld operations inherited from L. The larger eld L isthen said to be an extension eld of K. To simplify notation and terminology, one says that L / K (read as "L overK") is a eld extension to signify that L is an extension eld of K.If L is an extension of F which is in turn an extension ofK, then F is said to be an intermediate eld (or intermediateextension or subextension) of the eld extension L /K.Given a eld extension L /K and a subset S of L, the smallest subeld of L which contains K and S is denoted byK(S)i.e. K(S) is the eld generated by adjoining the elements of S to K. If S consists of only one element s, K(s) isa shorthand for K({s}). A eld extension of the form L = K(s) is called a simple extension and s is called a primitiveelement of the extension.Given a eld extension L /K, the larger eld L can be considered as a vector space over K. The elements of L arethe vectors and the elements of K are the scalars, with vector addition and scalar multiplication obtained fromthe corresponding eld operations. The dimension of this vector space is called the degree of the extension and isdenoted by [L : K].An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is nite orinnite the extension is called a nite extension or innite extension.

6.2 Caveats

The notation L /K is purely formal and does not imply the formation of a quotient ring or quotient group or any otherkind of division. Instead the slash expresses the word over. In some literature the notation L:K is used.It is often desirable to talk about eld extensions in situations where the small eld is not actually contained in thelarger one, but is naturally embedded. For this purpose, one abstractly denes a eld extension as an injective ringhomomorphism between two elds. Every non-zero ring homomorphism between elds is injective because elds donot possess nontrivial proper ideals, so eld extensions are precisely the morphisms in the category of elds.Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subelds.

10

6.3. EXAMPLES 11

6.3 ExamplesThe eld of complex numbers C is an extension eld of the eld of real numbers R, and R in turn is an extensioneld of the eld of rational numbersQ. Clearly then, C/Q is also a eld extension. We have [C : R] = 2 because {1,i}is a basis, so the extension C/R is nite. This is a simple extension because C=R( i ). [R : Q] = c (the cardinality ofthe continuum), so this extension is innite.The set Q(2) = {a + b2 | a, b Q} is an extension eld of Q, also clearly a simple extension. The degree is 2because {1, 2} can serve as a basis. Q(2, 3) = Q(2)( 3)={a + b3 | a, b Q(2)}={a + b2+ c3+ d6 | a,b,c,d Q} is an extension eld of both Q(2) and Q, of degree 2 and 4 respectively. Finite extensions of Q are alsocalled algebraic number elds and are important in number theory.Another extension eld of the rationals, quite dierent in avor, is the eld of p-adic numbersQp for a prime numberp.It is common to construct an extension eld of a given eld K as a quotient ring of the polynomial ring K[X] in orderto create a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 =1. Then the polynomial X2 + 1 is irreducible in K[X], consequently the ideal (X2 + 1) generated by this polynomialis maximal, and L = K[X]/(X2 + 1) is an extension eld of K which does contain an element whose square is 1(namely the residue class of X).By iterating the above construction, one can construct a splitting eld of any polynomial from K[X]. This is anextension eld L of K in which the given polynomial splits into a product of linear factors.If p is any prime number and n is a positive integer, we have a nite eld GF(pn) with pn elements; this is an extensioneld of the nite eld GF(p) = Z/pZ with p elements.Given a eld K, we can consider the eld K(X) of all rational functions in the variable X with coecients in K; theelements of K(X) are fractions of two polynomials over K, and indeed K(X) is the eld of fractions of the polynomialring K[X]. This eld of rational functions is an extension eld of K. This extension is innite.Given a Riemann surface M, the set of all meromorphic functions dened on M is a eld, denoted by C(M). It is anextension eld of C, if we identify every complex number with the corresponding constant function dened on M.Given an algebraic varietyV over some eldK, then the function eld ofV, consisting of the rational functions denedon V and denoted by K(V), is an extension eld of K.

6.4 Elementary propertiesIf L/K is a eld extension, then L and K share the same 0 and the same 1. The additive group (K,+) is a subgroup of(L,+), and the multiplicative group (K{0},) is a subgroup of (L{0},). In particular, if x is an element of K, thenits additive inverse x computed in K is the same as the additive inverse of x computed in L; the same is true formultiplicative inverses of non-zero elements of K.In particular then, the characteristics of L and K are the same.

6.5 Algebraic and transcendental elements and extensionsIf L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraicover K. Elements that are not algebraic are called transcendental. For example:

In C/R, i is algebraic because it is a root of x2 + 1. In R/Q, 2 + 3 is algebraic, because it is a root[1] of x4 10x2 + 1 In R/Q, e is transcendental because there is no polynomial with rational coecients that has e as a root (seetranscendental number)

In C/R, e is algebraic because it is the root of x e

The special case ofC/Q is especially important, and the names algebraic number and transcendental number are usedto describe the complex numbers that are algebraic and transcendental (respectively) over Q.

12 CHAPTER 6. FIELD EXTENSION

If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it issaid to be a transcendental extension.A subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coecients in Kexists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendencedegree of L/K. It is always possible to nd a set S, algebraically independent over K, such that L/K(S) is algebraic.Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to thetranscendence degree of the extension. An extension L/K is said to be purely transcendental if and only if thereexists a transcendence basis S of L/K such that L=K(S). Such an extension has the property that all elements of Lexcept those of K are transcendental over K, but, however, there are extensions with this property which are notpurely transcendentala class of such extensions take the form L/K where both L and K are algebraically closed.In addition, if L/K is purely transcendental and S is a transcendence basis of the extension, it doesn't necessarilyfollow that L=K(S). (For example, consider the extension Q(x,x)/Q, where x is transcendental over Q. The set {x}is algebraically independent since x is transcendental. Obviously, the extension Q(x,x)/Q(x) is algebraic, hence {x}is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x forx. But it is easy to see that {x} is a transcendence basis that generates Q(x,x)), so this extension is indeed purelytranscendental.)It can be shown that an extension is algebraic if and only if it is the union of its nite subextensions. In particular,every nite extension is algebraic. For example,

C/R and Q(2)/Q, being nite, are algebraic.

R/Q is transcendental, although not purely transcendental.

K(X)/K is purely transcendental.

A simple extension is nite if generated by an algebraic element, and purely transcendental if generated by a tran-scendental element. So

R/Q is not simple, as it is neither nite nor purely transcendental.

Every eld K has an algebraic closure; this is essentially the largest extension eld of K which is algebraic over K andwhich contains all roots of all polynomial equations with coecients in K. For example, C is the algebraic closure ofR.

6.6 Normal, separable and Galois extensionsAn algebraic extension L/K is called normal if every irreducible polynomial in K[X] that has a root in L completelyfactors into linear factors over L. Every algebraic extension F/K admits a normal closure L, which is an extension eldof F such that L/K is normal and which is minimal with this property.An algebraic extension L/K is called separable if the minimal polynomial of every element of L over K is separable,i.e., has no repeated roots in an algebraic closure over K. A Galois extension is a eld extension that is both normaland separable.A consequence of the primitive element theorem states that every nite separable extension has a primitive element(i.e. is simple).Given any eld extensionL/K, we can consider its automorphismgroupAut(L/K), consisting of all eld automorphisms: L L with (x) = x for all x in K. When the extension is Galois this automorphism group is called the Galoisgroup of the extension. Extensions whose Galois group is abelian are called abelian extensions.For a given eld extension L/K, one is often interested in the intermediate elds F (subelds of L that contain K).The signicance of Galois extensions and Galois groups is that they allow a complete description of the intermediateelds: there is a bijection between the intermediate elds and the subgroups of the Galois group, described by thefundamental theorem of Galois theory.

6.7. GENERALIZATIONS 13

6.7 GeneralizationsField extensions can be generalized to ring extensions which consist of a ring and one of its subrings. A closer non-commutative analog are central simple algebras (CSAs) ring extensions over a eld, which are simple algebra (nonon-trivial 2-sided ideals, just as for a eld) and where the center of the ring is exactly the eld. For example, the onlynite eld extension of the real numbers is the complex numbers, while the quaternions are a central simple algebraover the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be furthergeneralized to Azumaya algebras, where the base eld is replaced by a commutative local ring.

6.8 Extension of scalarsMain article: Extension of scalars

Given a eld extension, one can "extend scalars" on associated algebraic objects. For example, given a real vectorspace, one can produce a complex vector space via complexication. In addition to vector spaces, one can performextension of scalars for associative algebras dened over the eld, such as polynomials or group algebras and theassociated group representations. Extension of scalars of polynomials is often used implicitly, by just considering thecoecients as being elements of a larger eld, but may also be considered more formally. Extension of scalars hasnumerous applications, as discussed in extension of scalars: applications.

6.9 See also Field theory Glossary of eld theory Tower of elds Primary extension Regular extension

6.10 Notes[1] Wolfram|Alpha input: sqrt(2)+sqrt(3)". Retrieved 2010-06-14.

6.11 References Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised thirded.), New York: Springer-Verlag, ISBN 978-0-387-95385-4

6.12 External links Hazewinkel, Michiel, ed. (2001), Extension of a eld, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 7

Finite eld

In mathematics, a nite eld or Galois eld (so-named in honor of variste Galois) is a eld that contains a nitenumber of elements. As with any eld, a nite eld is a set on which the operations of multiplication, addition,subtraction and division are dened and satisfy certain basic rules. The most common examples of nite elds aregiven by the integers mod n when n is a prime number.The number of elements of a nite eld is called its order. A nite eld of order q exists if and only if the order q isa prime power pk (where p is a prime number and k is a positive integer). All elds of a given order are isomorphic.In a eld of order pk, adding p copies of any element always results in zero; that is, the characteristic of the eld is p.In a nite eld of order q, the polynomial Xq X has all q elements of the nite eld as roots. The non-zero elementsof a nite eld form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powersof a single element called a primitive element of the eld (in general there will be several primitive elements for agiven eld.)A eld has, by denition, a commutative multiplication operation. A more general algebraic structure that satisesall the other axioms of a eld but isn't required to have a commutative multiplication is called a division ring (orsometimes skeweld). A nite division ring is a nite eld by Wedderburns little theorem. This result shows that theniteness condition in the denition of a nite eld can have algebraic consequences.Finite elds are fundamental in a number of areas of mathematics and computer science, including number theory,algebraic geometry, Galois theory, nite geometry, cryptography and coding theory.

Commutative rings integral domains integrally closed domains unique factorization do-mains principal ideal domains Euclidean domains elds nite elds

7.1 Denitions, rst examples, and basic propertiesA nite eld is a nite set on which the four operations multiplication, addition, subtraction and division (excludingby zero) are dened, satisfying the rules of arithmetic known as the eld axioms. The simplest examples of niteelds are the prime elds: for each prime number p, the eld GF(p) (also denoted Z/pZ, Fp , or Fp) of order (that is,size) p is easily constructed as the integers modulo p.The elements of a prime eld may be represented by integers in the range 0, ..., p 1. The sum, the dierence andthe product are computed by taking the remainder by p of the integer result. The multiplicative inverse of an elementmay be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm Modular integers).Let F be a nite eld. For any element x in F and any integer n, let us denote by nx the sum of n copies of x. Theleast positive n such that n1 = 0 must exist and is prime; it is called the characteristic of the eld.If the characteristic of F is p, the operation (k; x) 7! k xmakes F a GF(p)-vector space. It follows that the numberof elements of F is pn.For every prime number p and every positive integer n, there are nite elds of order pn, and all these elds areisomorphic (see Existence and uniqueness below). One may therefore identify all elds of order pn, which aretherefore unambiguously denoted Fpn , Fpn or GF(pn), where the letters GF stand for Galois eld.[1]

14

7.2. EXISTENCE AND UNIQUENESS 15

The identity

(x+ y)p = xp + yp

is true (for every x and y) in a eld of characteristic p. (This follows from the fact that all, except the rst and the last,binomial coecients of the expansion of (x+ y)p are multiples of p).For every element x in the prime eld GF(p), one has xp = x (This is an immediate consequence of Fermats littletheorem, and this may be easily proved as follows: the equality is trivially true for x = 0 and x = 1; one obtains theresult for the other elements of GF(p) by applying the above identity to x and 1, where x successively takes the values1, 2, ..., p 1 modulo p.) This implies the equality

Xp X =Y

a2GF(p)(X a)

for polynomials over GF(p). More generally, every element in GF(pn) satises the polynomial equation xpn x = 0.Any nite eld extension of a nite eld is separable and simple. That is, if E is a nite eld and F is a subeld ofE, then E is obtained from F by adjoining a single element whose minimal polynomial is separable. To use a jargon,nite elds are perfect.

7.2 Existence and uniquenessLet q = pn be a prime power, and F be the splitting eld of the polynomial

P = Xq Xover the prime eld GF(p). This means that F is a nite eld of lowest order, in which P has q distinct roots (theroots are distinct, as the formal derivative of P is equal to 1). Above identity shows that the sum and the product oftwo roots of P are roots of P, as well as the multiplicative inverse of a root of P. In other word, the roots of P form aeld of order q, which is equal to F by the minimality of the splitting eld.The uniqueness up to isomorphism of splitting elds implies thus that all elds of order q are isomorphic.In summary, we have the following classication theorem rst proved in 1893 by E. H. Moore:[2]

The order of a nite eld is a prime power. For every prime power q there are elds of orderq, and they are all isomorphic. In these elds, every element satises

xq = x;

and the polynomial Xq X factors as

Xq X =Ya2F

(X a):

It follows that GF(pn) contains a subeld isomorphic to GF(pm) if and only if m is a divisor of n; in that case, thissubeld is unique. In fact, the polynomial Xpm X divides Xpn X if and only if m is a divisor of n.

7.3 Explicit construction of nite elds

7.3.1 Non-prime eldsGiven a prime power q = pn with p prime and n > 1, the eld GF(q) may be explicitly constructed in the followingway. One chooses rst an irreducible polynomial P in GF(p)[X] of degree n (such an irreducible polynomial alwaysexists). Then the quotient ring

16 CHAPTER 7. FINITE FIELD

GF(q) = GF(p)[X]/(P )

of the polynomial ring GF(p)[X] by the ideal generated by P is a eld of order q.More explicitly, the elements of GF(q) are the polynomials over GF(p) whose degree is strictly less than n. Theaddition and the subtraction are those of polynomials over GF(p). The product of two elements is the remainderof the Euclidean division by P of the product in GF(p)[X]. The multiplicative inverse of a non-zero element maybe computed with the extended Euclidean algorithm; see Extended Euclidean algorithm Simple algebraic eldextensions.Except in the construction of GF(4), there are several possible choices for P, which produce isomorphic results. Tosimplify the Euclidean division, for P one commonly chooses polynomials of the form

Xn + aX + b;

which make the needed Euclidean divisions very ecient. However, for some elds, typically in characteristic 2,irreducible polynomials of the formXn+ aX + bmay not exist. In characteristic 2, if the polynomial Xn + X + 1 isreducible, it is recommended to choose Xn + Xk + 1 with the lowest possible k that makes the polynomial irreducible.If all these trinomials are reducible, one chooses pentanomials Xn + Xa + Xb + Xc + 1, as polynomials of degreegreater than 1, with an even number of terms, are never irreducible in characteristic 2, having 1 as a root.[3]

In the next sections, we will show how this general construction method works for small nite elds.

7.3.2 Field with four elementsOver GF(2), there is only one irreducible polynomial of degree 2:

X2 +X + 1

Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and

GF(4) = GF(2)[X]/(X2 +X + 1):

If one denotes a a root of this polynomial in GF(4), the tables of the operations in GF(4) are the following. Thereis no table for subtraction, as, in every eld of characteristic 2, subtraction is identical to addition. In the third table,for the division of x by y, x must be read on the left, and y on the top.

7.3.3 GF(p2) for an odd prime pFor applying above general construction of nite elds in the case of GF(p2), one has to nd an irreducible polynomialof degree 2. For p = 2, this has been done in the preceding section. If p is an odd prime, there are always irreduciblepolynomials of the form X2 r, with r in GF(p).More precisely, the polynomial X2 r is irreducible over GF(p) if and only if r is a quadratic non-residue modulop (this is almost the denition of a quadratic non-residue). There are p12 quadratic non-residues modulo p. Forexample, 2 is a quadratic non-residue for p = 3, 5, 11, 13, ..., and 3 is a quadratic non-residue for p = 5, 7, 17, .... Ifp 3 mod 4, that is p = 3, 7, 11, 19, ..., one may choose 1 p 1 as a quadratic non-residue, which allows us tohave a very simple irreducible polynomial X2 + 1.Having chosen a quadratic non-residue r, let be a symbolic square root of r, that is a symbol which has the property2 = r, in the same way as the complex number i is a symbolic square root of 1. Then, the elements of GF(p2) areall the linear expressions

a+ b;

7.3. EXPLICIT CONSTRUCTION OF FINITE FIELDS 17

with a and b in GF(p). The operations on GF(p2) are dened as follows (the operations between elements of GF(p)represented by Latin letters are the operations in GF(p)):

(a+ b) = a+ (b)(a+ b) + (c+ d) = (a+ c) + (b+ d)

(a+ b)(c+ d) = (ac+ rbd) + (ad+ bc)

(a+ b)1 = a(a2 rb2)1 + (b)(a2 rb2)1

7.3.4 GF(8) and GF(27)The polynomial

X3 X 1is irreducible over GF(2) and GF(3), that is, it is irreducible modulo 2 and 3 (to show this it suce to show that it hasno root in GF(2) nor in GF(3)). It follows that the elements of GF(8) and GF(27) may be represented by expressions

a+ b+ c2;

where a, b, c are elements of GF(2) or GF(3) (respectively), and is a symbol such that

3 = + 1:

The addition, additive inverse and multiplication on GF(8) and GF(27) may thus be dened as follows; in followingformulas, the operations between elements of GF(2) or GF(3), represented by Latin letters are the operations in GF(2)or GF(3), respectively:

(a+ b+ c2) = a+ (b)+ (c)2 (ForGF (8); identity) the is operation this(a+ b+ c2) + (d+ e+ f2) = (a+ d) + (b+ e)+ (c+ f)2

(a+ b+ c2)(d+ e+ f2) = (ad+ bf + ce) + (ae+ bd+ bf + ce+ cf)+ (af + be+ cd+ cf)2

7.3.5 GF(16)The polynomial

X4 +X + 1

is irreducible over GF(2), that is, it is irreducible modulo 2. It follows that the elements of GF(16) may be representedby expressions

a+ b+ c2 + d3;

where a, b, c, d are either 0 or 1 (elements of GF(2)), and is a symbol such that

4 = + 1:

As the characteristic of GF(2) is 2, each element is its additive inverse in GF(16). The addition and multiplicationon GF(16) may be dened as follows; in following formulas, the operations between elements of GF(2), representedby Latin letters are the operations in GF(2).


(a+ b+ c2 + d3) + (e+ f+ g2 + h3) = (a+ e) + (b+ f)+ (c+ g)2 + (d+ h)3

(a+ b+ c2 + d3)(e+ f+ g2 + h3) = (ae+ bh+ cg + df) + (af + be+ bh+ cg + df + ch+ dg) +

(ag + bf + ce+ ch+ dg + dh)2 + (ah+ bg + cf + de+ dh)3

7.4 Multiplicative structureThe set of non-zero elements in GF(q) is an Abelian group under the multiplication, of order q 1. By Lagrangestheorem, there exists a divisor k of q 1 such that xk = 1 for every non-zero x in GF(q). As the equation Xk = 1 has atmost k solutions in any eld, q 1 is the lowest possible value for k. The structure theorem of nite Abelian groupsimplies that this multiplicative group is cyclic, that all non-zero elements are powers of single element. In summary:

The multiplicative group of the non-zero elements in GF(q) is cyclic, and there exist an element a, suchthat the q 1 non-zero elements of GF(q) are a, a2, ..., aq2, aq1 = 1.

Such an element a is called a primitive element. Unless q = 2, 3, the primitive element is not unique. The number ofprimitive elements is (q 1) where is Eulers totient function.Above result implies that xq = x for every x in GF(q). The particular case where q is prime is Fermats little theorem.

7.4.1 Discrete logarithmIf a is a primitive element in GF(q), then for any non-zero element x in F, there is a unique integer n with 0 n q 2 such that

x = an.

This integer n is called the discrete logarithm of x to the base a.While the computation of an is rather easy, by using, for example, exponentiation by squaring, the reciprocal oper-ation, the computation of the discrete logarithm is dicult. This has been used in various cryptographic protocols,see Discrete logarithm for details.When the nonzero elements of GF(q) are represented by their discrete logarithms, multiplication and division areeasy, as they reduce to addition and subtraction modulo q 1. However, addition amounts to computing the discretelogarithm of am + an. The identity

am + an = an(amn + 1)

allows one to solve this problem by constructing the table of the discrete logarithms of an + 1, called Zechs logarithms,for n = 0, ..., q 2 (it is convenient to dene the discrete logarithm of zero as being ).Zechs logarithms are useful for large computations, such as linear algebra over medium-sized elds, that is, elds thatare suciently large for making natural algorithms inecient, but not too large, as one has to pre-compute a table ofthe same size as the order of the eld.

7.4.2 Roots of unityEvery nonzero element of a nite eld is a root of unity, as xq1 = 1 for every nonzero element of GF(q).If n is a positive integer, a nth primitive root of unity is a solution of the equation xn = 1 that is not a solution of theequation xm = 1 for any positive integer m < n. If a is a nth primitive root of unity in a eld F, then F contains all then roots of unity, which are 1, a, a2, ..., an1.The eld GF(q) contains a nth primitive root of unity if and only if n is a divisor of q 1; if n is a divisor of q 1,then the number of primitive nth roots of unity in GF(q) is (n) (Eulers totient function). The number of nth rootsof unity in GF(q) is gcd(n, q 1).

7.4. MULTIPLICATIVE STRUCTURE 19

In a eld of characteristic p, every (np)th root of unity is also a nth root of unity. It follows that primitive (np)th rootsof unity never exist in a eld of characteristic p.On the other hand, if n is coprime to p, the roots of the nth cyclotomic polynomial are distinct in every eld ofcharacteristic p, as this polynomial is a divisor of Xn 1, which has 1 as formal derivative. It follows that the nthcyclotomic polynomial factors over GF(p) into distinct irreducible polynomials that have all the same degree, say d,and that GF(pd) is the smallest eld of characteristic p that contains the nth primitive roots of unity.

7.4.3 Example

The eld GF(64) has several interesting properties that smaller elds do not share. Specically, it has two subeldssuch that neither is a subeld of the other, not all generators (elements having a minimal polynomial of degree 6 overGF(2)) are primitive elements, and the primitive elements are not all conjugate under the Galois group.The order of this eld being 26, and the divisors of 6 being 1, 2, 3, 6, the subelds of GF(64) are GF(2), GF(22) =GF(4), GF(23) = GF(8), and GF(64) itself. As 2 and 3 are coprime, the intersection of GF(4) and GF(8) in GF(64)is the prime eld GF(2).The union of GF(4) and GF(8) has thus 10 elements. The remaining 54 elements of GF(64) generate GF(64) in thesense that no other subeld contains any of them. It follows that they are roots of irreducible polynomials of degree6 over GF(2). This implies that, over GF(2), there are exactly 9 = 54/6 irreducible monic polynomials of degree 6.This may be veried by factoring X64 X over GF(2).The elements of GF(64) are primitive nth roots of unity for some n dividing 63. As the 3rd and the 7th roots of unitybelong to GF(4) and GF(8), respectively, the 54 generators are primitive nth roots of unity for some n in {9, 21, 63}.Eulers totient function shows that there are 6 primitive 9th roots of unity, 12 primitive 21st roots of unity, and 36primitive 63rd roots of unity. Summing these numbers, one nds again 54 elements.By factoring the cyclotomic polynomials over GF(2), one nds that:

The six primitive 9th roots of unity are roots of

X6 +X3 + 1;

and are all conjugate under the action of the Galois group.

The twelve primitive 21st roots of unity are roots of

(X6 +X4 +X2 +X + 1)(X6 +X5 +X4 +X2 + 1):

They form two orbits under the action of the Galois group. As the two factors are reciprocal to eachother, a root and its (multiplicative) inverse do not belong to the same orbit.

The 36 primitive elements of GF(64) are the roots of

(X6+X4+X3+X+1)(X6+X+1)(X6+X5+1)(X6+X5+X3+X2+1)(X6+X5+X2+X+1)(X6+X5+X4+X+1);

They split into 6 orbits of 6 elements under the action of the Galois group.

This shows that the best choice to construct GF(64) is to dene it as GF(2)[X]/(X6 + X + 1). In fact, this generatoris a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.


7.5 Frobenius automorphism and Galois theoryIn this section, p is a prime number, and q = pn is a power of p.In GF(q), the identity (x+ y)p = xp + yp implies that the map

' : x 7! xp

is a GF(p)-linear endomorphism and a eld automorphism of GF(q), which xes every element of the subeld GF(p).It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.Denoting by 'k the composition of ' with itself, k times, we have

'k : x 7! xpk :

It has been shown in the preceding section that 'n is the identity. For 0 < k < n, the automorphism 'k is not theidentity, as, otherwise, the polynomial

Xpk X

would have more than pk roots.There are no other GF(p)-automorphisms of GF(q). In other words, GF(pn) has exactly n GF(p)-automorphisms,which are

Id = '0; '; '2; : : : ; 'n1:

In terms of Galois theory, this means that GF(pn) is a Galois extension of GF(p), which has a cyclic Galois group.The fact that the Frobenius map is surjective implies that every nite eld is perfect.

7.6 Polynomial factorizationMain article: Factorization of polynomials over nite elds

If F is a nite eld, a non-constant monic polynomial with coecients in F is irreducible over F, if it is not the productof two non-constant monic polynomials, with coecients in F.As every polynomial ring over a eld is a unique factorization domain, every monic polynomial over a nite eld maybe factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials.There are ecient algorithms for testing polynomial irreducibility and factoring polynomials over nite eld. Theyare a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, everycomputer algebra system has functions for factoring polynomials over nite elds, or, at least, over nite prime elds.

7.6.1 Irreducible polynomials of a given degreeThe polynomial

Xq X

factors into linear factors over a eld of order q. More precisely, this polynomial is the product of all monic polyno-mials of degree one over a eld of order q.

7.7. APPLICATIONS 21

This implies that, if q = pn that Xq X is the product of all monic irreducible polynomials over GF(p), whose degreedivides n. In fact, if P is an irreducible factor over GF(p) of Xq X, its degree divides n, as its splitting eld iscontained in GF(pn). Conversely, if P is an irreducible monic polynomial over GF(p) of degree d dividing n, itdenes a eld extension of degree d, which is contained in GF(pn), and all roots of P belong to GF(pn), and are rootsof Xq X; thus P divides Xq X. As Xq X does not have any multiple factor, it is thus the product of all theirreducible monic polynomials that divide it.This property is used to compute the product of the irreducible factors of each degree of polynomials over GF(p);see Distinct degree factorization.

7.6.2 Number of monic irreducible polynomials of a given degree over a nite eldThe number N(q,n) of monic irreducible polynomials of degree n over GF(q) is given by[4]

N(q; n) =1

n

Xdjn

(d)qnd ;

where is the Mbius function. This formula is almost a direct consequence of above property of Xq X.By the above formula, the number of irreducible (not necessarily monic) polynomials of degree n over GF(q) is (q 1)N(q, n).A (slightly simpler) lower bound for N(q, n) is

N(q; n) 1n

0@qn Xpjn; pprime

qnp

1A :One may easily deduce that, for every q and every n, there is at least one irreducible polynomial of degree n overGF(q). This lower bound is sharp for q = n = 2.

7.7 ApplicationsIn cryptography, the diculty of the discrete logarithm problem in nite elds or in elliptic curves is the basis ofseveral widely used protocols, such as the DieHellman protocol. For example, in 2014, the secure connection toWikipedia involves the elliptic curve DieHellman protocol (ECDHE) over a large nite eld.[5] In coding theory,many codes are constructed as subspaces of vector spaces over nite elds.Finite elds are widely used in number theory, as many problems over the integers may be solved by reducing themmodulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization andlinear algebra over the eld of rational numbers proceed by reduction modulo one or several primes, and then recon-struction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.Similarly many theoretical problems in number theory can be solved by considering their reductions modulo someor all prime numbers. See, for example, Hasse principle. Many recent developments of algebraic geometry weremotivated by the need to enlarge the power of these modular methods. Wiles proof of Fermats Last Theorem is anexample of a deep result involving many mathematical tools, including nite elds.

7.8 Extensions

7.8.1 Algebraic closureA nite eld F is not algebraically closed. To demonstrate this, consider the polynomial

f(T ) = 1 +Y2F

(T );


which has no roots in F, since f () = 1 for all in F.The direct limit of the system:

{Fp, Fp2, ..., Fpn, ...},

with inclusion, is an innite eld. It is the algebraic closure of all the elds in the system, and is denoted by: Fp .The inclusions commute with the Frobenius map, as it is dened the same way on each eld (x x p ), so theFrobenius map denes an automorphism of Fp , which carries all subelds back to themselves. In fact Fpn can berecovered as the xed points of the nth iterate of the Frobenius map.However unlike the case of nite elds, the Frobenius automorphism on Fp has innite order, and it does not generatethe full group of automorphisms of this eld. That is, there are automorphisms of Fp which are not a power of theFrobenius map. However, the group generated by the Frobenius map is a dense subgroup of the automorphism groupin the Krull topology. Algebraically, this corresponds to the additive group Z being dense in the pronite integers(direct product of the p-adic integers over all primes p, with the product topology).If we actually construct our nite elds in such a fashion that Fpn is contained in Fpm whenever n divides m, then thisdirect limit can be constructed as the union of all these elds. Even if we do not construct our elds this way, we canstill speak of the algebraic closure, but some more delicacy is required in its construction.

7.8.2 Wedderburns little theoremA division ring is a generalization of eld. Division rings are not assumed to be commutative. There are no non-commutative nite division rings: Wedderburns little theorem states that all nite division rings are commutative,hence nite elds. The result holds even if we relax associativity and consider alternative rings, by the ArtinZorntheorem.

7.9 See also Quasi-nite eld Field with one element Finite eld arithmetic Trigonometry in Galois elds Finite ring Finite group elementary abelian group Hamming space

7.10 Notes[1] This notation was introduced by E. H. Moore in an address given in 1893 at the International Mathematical Congress held

in Chicago Mullen & Panario 2013, p. 10.

[2] Moore, E. H. (1896), A doubly-innite system of simple groups, in E. H. Moore, et. al., Mathematical Papers Read atthe International Mathematics Congress Held in Connection with the Worlds Columbian Exposition, Macmillan & Co., pp.208242

[3] NIST, Recommended Elliptic Curves for Government Use, page 3

[4] Jacobson 2009, 4.13

[5] This can be veried by looking at the information on the page provided by the browser.

7.11. REFERENCES 23

7.11 References Jacobson, Nathan (2009) [1985], Basic algebra I (Second ed.), Dover Publications, ISBN 978-0-486-47189-1 L. Mullen, Garry; Mummert, Carl (2007), Finite Fields and Applications I, Student Mathematical Library(AMS), ISBN 978-0-8218-4418-2

Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6 Lidl, Rudolf; Niederreiter, Harald (1997), Finite Fields (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4

7.12 External links Finite Fields at Wolfram research.

Chapter 8

Galois extension

In mathematics, a Galois extension is an algebraic eld extension E/F that is normal and separable; or equivalently,E/F is algebraic, and the eld xed by the automorphism group Aut(E/F) is precisely the base eld F. The signicanceof being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galoistheory. [1]

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given eld, and G is a nite groupof automorphisms of E with xed eld F, then E/F is a Galois extension.

8.1 Characterization of Galois extensionsAn important theorem of Emil Artin states that for a nite extensionE/F, each of the following statements is equivalentto the statement that E/F is Galois:

E/F is a normal extension and a separable extension. E is a splitting eld of a separable polynomial with coecients in F. |Aut(E/F)| = [E:F], that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

Every irreducible polynomial in F[x] with at least one root in E splits over E and is separable. |Aut(E/F)| [E:F], that is, the number of automorphisms is at least the degree of the extension. F is the xed eld of a subgroup of Aut(E). F is the xed eld of Aut(E/F). There is a one-to-one correspondence between subelds of E/F and subgroups of Aut(E/F).

8.2 ExamplesThere are two basic ways to construct examples of Galois extensions.

Take any eld E, any subgroup of Aut(E), and let F be the xed eld. Take any eld F, any separable polynomial in F[x], and let E be its splitting eld.

Adjoining to the rational number eld the square root of 2 gives a Galois extension, while adjoining the cube root of2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The rst

24

8.3. REFERENCES 25

of them is the splitting eld of x2 2; the second has normal closure that includes the complex cube roots of unity,and so is not a splitting eld. In fact, it has no automorphism other than the identity, because it is contained in thereal numbers and x3 2 has just one real root. For more detailed examples, see the page on the fundamental theoremof Galois theoryAn algebraic closure K of an arbitrary eldK is Galois overK if and only ifK is a perfect eld.

8.3 References[1] See the article Galois group for denitions of some of these terms and some examples.

8.4 See also Artin, Emil (1998). Galois Theory. Edited and with a supplemental chapter by Arthur N. Milgram. Mineola,NY: Dover Publications. ISBN 0-486-62342-4. MR 1616156.

Bewersdor, Jrg (2006). Galois theory for beginners. Student Mathematical Library 35. Translated fromthe second German (2004) edition by David Kramer. American Mathematical Society. ISBN 0-8218-3817-2.MR 2251389.

Edwards, HaroldM. (1984). Galois Theory. Graduate Texts in Mathematics 101. New York: Springer-Verlag.ISBN 0-387-90980-X. MR 0743418. (Galois original paper, with extensive background and commentary.)

Funkhouser, H. Gray (1930). A short account of the history of symmetric functions of roots of equations.American Mathematical Monthly (The American Mathematical Monthly, Vol. 37, No. 7) 37 (7): 357365.doi:10.2307/2299273. JSTOR 2299273.

Hazewinkel, Michiel, ed. (2001), Galois theory, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Jacobson, Nathan (1985). Basic Algebra I (2nd ed.). W.H. Freeman and Company. ISBN 0-7167-1480-9.(Chapter 4 gives an introduction to the eld-theoretic approach to Galois theory.)

Janelidze, G.; Borceux, Francis (2001). Galois theories. Cambridge University Press. ISBN 978-0-521-80309-0. (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leadingto Galois groupoids.)

Lang, Serge (1994). Algebraic Number Theory. Graduate Texts in Mathematics 110 (Second ed.). Berlin,New York: Springer-Verlag. doi:10.1007/978-1-4612-0853-2. ISBN 978-0-387-94225-4. MR 1282723.

Postnikov, Mikhail Mikhalovich (2004). Foundations of Galois Theory. With a foreword by P. J. Hilton.Reprint of the 1962 edition. Translated from the 1960 Russian original by Ann Swinfen. Dover Publications.ISBN 0-486-43518-0. MR 2043554.

Rotman, Joseph (1998). Galois Theory (Second ed.). Springer. doi:10.1007/978-1-4612-0617-0. ISBN 0-387-98541-7. MR 1645586.

Vlklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge Studies in Advanced Mathe-matics 53. Cambridge University Press. doi:10.1017/CBO9780511471117. ISBN 978-0-521-56280-5. MR1405612.

van der Waerden, Bartel Leendert (1931). Moderne Algebra (in German). Berlin: Springer.. English trans-lation (of 2nd revised edition): Modern algebra. New York: Frederick Ungar. 1949. (Later republished inEnglish by Springer under the title Algebra.)

Pop, Florian (2001). "(Some) New Trends in Galois Theory and Arithmetic (PDF).

Chapter 9

Normal extension

In abstract algebra, an algebraic eld extension L/K is said to be normal if L is the splitting eld of a family ofpolynomials in K[X]. Bourbaki calls such an extension a quasi-Galois extension.

9.1 Equivalent properties and examplesThe normality of L/K is equivalent to either of the following properties. LetKa be an algebraic closure ofK containingL.

Every embedding of L in Ka that restricts to the identity on K, satises (L) = L. In other words, is anautomorphism of L over K.

Every irreducible polynomial in K[X] that has one root in L, has all of its roots in L, that is, it decomposes intolinear factors in L[X]. (One says that the polynomial splits in L.)

If L is a nite extension of K that is separable (for example, this is automatically satised if K is nite or has charac-teristic zero) then the following property is also equivalent:

There exists an irreducible polynomial whose roots, together with the elements of K, generate L. (One says thatL is the splitting eld for the polynomial.)

For example, Q(p2) is a normal extension of Q , since it is a splitting eld of x2 2. On the other hand, Q( 3

p2) is

not a normal extension of Q since the irreducible polynomial x3 2 has one root in it (namely, 3p2 ), but not all of

them (it does not have the non-real cubic roots of 2).The fact thatQ( 3

p2) is not a normal extension ofQ can also be seen using the rst of the three properties above. The

eld A of algebraic numbers is an algebraic closure of Q containing Q( 3p2) . On the other hand

Q( 3p2) = fa+ b 3

p2 + c

3p4 2 A j a; b; c 2 Qg

and, if is one of the two non-real cubic roots of 2, then the map

: Q( 3p2) ! A

a+ b 3p2 + c 3

p4 7! a+ b! 3p2 + c!2 3p4

is an embedding ofQ( 3p2) inA whose restriction toQ is the identity. However, is not an automorphism ofQ( 3

p2)

.For any prime p, the extension Q( p

p2; p) is normal of degree p(p 1). It is a splitting eld of xp 2. Here p

denotes any pth primitive root of unity. The eld Q( 3p2; 3) is the normal closure (see below) of Q( 3

p2) .

26

9.2. OTHER PROPERTIES 27

9.2 Other propertiesLet L be an extension of a eld K. Then:

If L is a normal extension of K and if E is an intermediate extension (i.e., L E K), then L is a normalextension of E.

If E and F are normal extensions of K contained in L, then the compositum EF and E F are also normalextensions of K.

9.3 Normal closureIf K is a eld and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is anormal extension of K. Furthermore, up to isomorphism there is only one such extension which is minimal, i.e. suchthat the only subeld ofM which contains L and which is a normal extension of K isM itself. This extension is calledthe normal closure of the extension L of K.If L is a nite extension of K, then its normal closure is also a nite extension.

9.4 See also Galois extension Normal basis

9.5 References Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787

Chapter 10

Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.More explicitly, if R and S are rings, then a ring homomorphism is a function f : R S such that[1][2][3][4][5][6]

f(a + b) = f(a) + f(b) for all a and b in R f(ab) = f(a) f(b) for all a and b in R f(1R) = 1S.

(Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitlythat they too are respected, because these conditions are consequences of the three conditions above. On the otherhand, neglecting to include the condition f(1R) = 1S would cause several of the properties below to fail.)If R and S are rngs (also known as pseudo-rings, or non-unital rings), then the natural notion[7] is that of a rng homo-morphism, dened as above except without the third condition f(1R) = 1S. It is possible to have a rng homomorphismbetween (unital) rings that is not a ring homomorphism.The composition of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings formsa category with ring homomorphisms as the morphisms (cf. the category of rings). In particular, one obtains thenotions of ring endomorphism, ring isomorphism, and ring automorphism.

10.1 PropertiesLet f : R S be a ring homomorphism. Then, directly from these denitions, one can deduce:

f(0R) = 0S. f(a) = f(a) for all a in R. For any unit element a in R, f(a) is a unit element such that f(a1) = f(a)1. In particular, f induces a grouphomomorphism from the (multiplicative) group of units of R to the (multiplicative) group of units of S (or ofim(f)).

The image of f, denoted im(f), is a subring of S. The kernel of f, dened as ker(f) = {a in R : f(a) = 0}, is an ideal in R. Every ideal in a commutative ring Rarises from some ring homomorphism in this way.

The homomorphism f is injective if and only if ker(f) = {0}. If f is bijective, then its inverse f1 is also a ring homomorphism. In this case, f is called an isomorphism,and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot bedistinguished.

28

10.2. EXAMPLES 29

If there exists a ring homomorphism f : RS then the characteristic of S divides the characteristic of R. Thiscan sometimes be used to show that between certain rings R and S, no ring homomorphisms R S can exist.

If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ringhomomorphism f : R S induces a ring homomorphism fp : Rp Sp.

If R is a eld and S is not the zero ring, then f is injective. If both R and S are elds, then im(f) is a subeld of S, so S can be viewed as a eld extension of R. If R and S are commutative and P is a prime ideal of S then f1(P) is a prime ideal of R. If R and S are commutative and S is an integral domain, then ker(f) is a prime ideal of R. If R and S are commutative, S is a eld, and f is surjective, then ker(f) is a maximal ideal of R. If f is surjective, P is prime (maximal) ideal in R and ker(f) P, then f(P) is prime (maximal) ideal in S.

Moreover,

The composition of ring homomorphisms is a ring homomorphism. The identity map is a ring homomorphism (but not the zero map). Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings. For every ring R, there is a unique ring homomorphism Z R. This says that the ring of integers is an initialobject in the category of rings.

For every ring R, there is a unique ring homomorphism R 0, where 0 denotes the zero ring (the ring whoseonly element is zero). This says that the zero ring is a terminal object in the category of rings.

10.2 Examples The function f : Z Zn, dened by f(a) = [a]n = amod n is a surjective ring homomorphism with kernel nZ(see modular arithmetic).

The function f : Z6 Z6 dened by f([a]6) = [4a]6 is a rng homomorphism (and rng endomorphism), withkernel 3Z6 and image 2Z6 (which is isomorphic to Z3).

There is no ring homomorphism Zn Z for n 1. The complex conjugation CC is a ring homomorphism (in fact, an example of a ring automorphism.) If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the zero ring.(Otherwise it fails to map 1R to 1S.) On the other hand, the zero function is always a rng homomorphism.

If R[X] denotes the ring of all polynomials in the variable X with coecients in the real numbers R, and Cdenotes the complex numbers, then the function f : R[X] C dened by f(p) = p(i) (substitute the imaginaryunit i for the variable X in the polynomial p) is a surjective ring homomorphism. The kernel of f consists ofall polynomials in R[X] which are divisible by X2 + 1.

If f : R S is a ring homomorphism between the rings R and S, then f induces a ring homomorphism betweenthe matrix rings Mn(R) Mn(S).

10.3 The category of ringsMain article: Category of rings

30 CHAPTER 10. RING HOMOMORPHISM

10.3.1 Endomorphisms, isomorphisms, and automorphisms A ring endomorphism is a ring homomorphism from a ring to itself. A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism.One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function onthe underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are calledisomorphic. Isomorphic rings dier only by a relabeling of elements. Example: Up to isomorphism, there arefour rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that everyother ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are elevenrngs of order 4.

A ring automorphism is a ring isomorphism from a ring to itself.

10.3.2 Monomorphisms and epimorphismsInjective ring homomorphisms are identical to monomorphisms in the category of rings: If f : R S is a monomor-phism that is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2from Z[x] to R that map x to r1 and r2, respectively; f g1 and f g2 are identical, but since f is a monomorphismthis is impossible.However, surjective ring homomorphisms are vastly dierent from epimorphisms in the category of rings. For ex-ample, the inclusion Z Q is a ring epimorphism, but not a surjection. However, they are exactly the same as thestrong epimorphisms.

10.4 Notes[1] Artin, p. 353[2] Atiyah and Macdonald, p. 2[3] Bourbaki, p. 102[4] Eisenbud, p. 12[5] Jacobson, p. 103[6] Lang, p. 88[7] Hazewinkel et al. (2004), p. 3. Warning: They use the word ring to mean rng.

10.5 References Michael Artin, Algebra, Prentice-Hall, 1991. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969. N. Bourbaki, Algebra I, Chapters 1-3, 1998. David Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer, 1995. Michiel Hazewinkel, Nadiya Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1.2004. Springer, 2004. ISBN 1-4020-2690-0

Nathan Jacobson, Basic algebra I, 2nd edition, 1985. Serge Lang, Algebra 3rd ed., Springer, 2002.

10.6 See also change of rings

Chapter 11

Separable extension

In the subeld of algebra named eld theory, a separable extension is an algebraic eld extension E F such thatfor every 2 E , the minimal polynomial of over F is a separable polynomial (i.e., has distinct roots; see below forthe denition in this context).[1] Otherwise, the extension is called inseparable. There are other equivalent denitionsof the notion of a separable algebraic extension, and these are outlined later in the article.The importance of separable extensions lies in the fundamental role they play in Galois theory in nite characteristic.More specically, a nite degree eld extension is Galois if and only if it is both normal and separable.[2] Sincealgebraic extensions of elds of characteristic zero, and of nite elds, are separable, separability is not an obstaclein most applications of Galois theory.[3][4] For instance, every algebraic (in particular, nite degree) extension of theeld of rational numbers is necessarily separable.Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class ofpurely inseparable extensions, also occurs quite naturally. An algebraic extension E F is a purely inseparableextension if and only if for every 2 E nF , the minimal polynomial of over F is not a separable polynomial (i.e.,does not have distinct roots).[5] For a eld F to possess a non-trivial purely inseparable extension, it must necessarilybe an innite eld of prime characteristic (i.e. specically, imperfect), since any algebraic extension of a perfect eldis necessarily separable.[3]

11.1 Informal discussion

An arbitrary polynomial f with coecients in some eld F is said to have distinct roots if and only if it has deg(f)roots in some extension eld E F . For instance, the polynomial g(X)=X2+1 with real coecients has preciselydeg(g)=2 roots in the complex plane; namely the imaginary unit i, and its additive inverse i, and hence does havedistinct roots. On the other hand, the polynomial h(X)=(X2)2 with real coecients does not have distinct roots; only2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg(h)=2 roots.To test if a polynomial has distinct roots, it is not necessary to consider explicitly any eld extension nor to compute theroots: a polynomial has distinct roots if and only if the greatest common divisor of the polynomial and its derivativeis a constant. For instance, the polynomial g(X)=X2+1 in the above paragraph, has 2X as derivative, and, over a eldof characteristic dierent of 2, we have g(X) - (1/2 X) 2X = 1, which proves, by Bzouts identity, that the greatestcommon divisor is a constant. On the other hand, over a eld where 2=0, the greatest common divisor is g, and wehave g(X) = (X+1)2 has 1=1 as double root. On the other hand, the polynomial h does not have distinct roots,whichever is the eld of the coecients, and indeed, h(X)=(X2)2, its derivative is 2 (X2) and divides it, and hencedoes have a factor of the form (X )2 for = 2 ).Although an arbitrary polynomial with rational or real coecients may not have distinct roots, it is natural to ask atthis stage whether or not there exists an irreducible polynomial with rational or real coecients that does not havedistinct roots. The polynomial h(X)=(X2)2 does not have distinct roots but it is not irreducible as it has a non-trivialfactor (X2). In fact, it is true that there is no irreducible polynomial with rational or real coecients that does nothave distinct roots; in the language of eld theory, every algebraic extension of Q or R is separable and hence bothof these elds are perfect.

31

32 CHAPTER 11. SEPARABLE EXTENSION

11.2 Separable and inseparable polynomialsA polynomial f in F[X] is a separable polynomial if and only if every irreducible factor of f in F[X] has distinctroots.[6] The separability of a polynomial depends on the eld in which its coecients are considered to lie; forinstance, if g is an inseparable polynomial in F[X], and one considers a splitting eld, E, for g over F, g is necessarilyseparable in E[X] since an arbitrary irreducible factor of g in E[X] is linear and hence has distinct roots.[1] Despitethis, a separable polynomial h in F[X] must necessarily be separable over every extension eld of F.[7]

Let f in F[X] be an irreducible polynomial and f ' its formal derivative. Then the following are equivalent conditionsfor f to be separable; that is, to have distinct roots:

If E F and 2 E , then (X )2 does not divide f in E[X].[8]

There existsK F such that f has deg(f) roots in K.[8]

f and f ' do not have a common root in any extension eld of F.[9]

f ' is not the zero polynomial.[10]

By the last condition above, if an irreducible polynomial does not have distinct roots, its derivative must be zero.Since the formal derivative of a positive degree polynomial can be zero only if the eld has prime characteristic,for an irreducible polynomial to not have distinct roots its coecients must lie in a eld of prime characteristic.More generally, if an irreducible (non-zero) polynomial f in F[X] does not have distinct roots, not only must t

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