Field extensionsFrom 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 fields . . . . . . . . . . . . . . . . . . . . . . . . . 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 field extension 64.1 Definition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 The multiplicativity formula for degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.2.1 Proof of the multiplicativity formula in the finite case . . . . . . . . . . . . . . . . . . . . 74.2.2 Proof of the formula in the infinite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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

5 Dual basis in a field extension 9

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

i

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 Galois extension 147.1 Characterization of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Normal extension 168.1 Equivalent properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Normal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9 Separable extension 189.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.5 The definition of separable non-algebraic extension fields . . . . . . . . . . . . . . . . . . . . . . 209.6 Differential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10 Simple extension 2310.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2 Structure of simple extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

11 Tower of fields 2511.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS iii

11.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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 finite extension of a finite field is a cyclic extension. The development of class field theory has provided detailedinformation about abelian extensions of number fields, function fields of algebraic curves over finite fields, and localfields.There are two slightly different concepts of cyclotomic extensions: these can mean either extensions formed byadjoining roots of unity, or subextensions of such extensions. The cyclotomic fields are examples. Any cyclotomicextension (for either definition) is abelian.If a field 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 field of rational numbers, an extension is abelian if and only if it is a subfield of a fieldobtained by adjoining a root of unity.There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space:abelian covers are classified by its abelianisation which relates directly to the first homology group.

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

1

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Chapter 2

Algebraic closure

For other uses, see Closure (disambiguation).

In mathematics, particularly abstract algebra, an algebraic closure of a field 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 field has an algebraic closure,[1][2][3] and that the algebraic closureof a field K is unique up to an isomorphism that fixes 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 field 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 field containingK, because ifM is any algebraically closed field containing K, then the elements ofM that are algebraic over K forman algebraic closure of K.The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.[3]

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

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

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

For a finite field of prime power order q, the algebraic closure is a countably infinite field that contains a copyof the field 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 fields

Let S = {f| } be the set of all monic irreducible polynomials in K[x]. For each f 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 and all i degree(f) . Write

f d

i=1

(x u,i) =d1j=0

r,j xj R[x]

2

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2.3. SEPARABLE CLOSURE 3

with r,j 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 field of S over K.

2.3 Separable closure

An 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 finite separable extensions of Ksep, of degree > 1. Saying this anotherway, K is contained in a separably-closed algebraic extension field. It is essentially unique (up to isomorphism).[5]

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

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

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

2.4 See also Algebraically closed field

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, Infinite 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 fields (Corrected reprint of the 2nd ed.). New York: DoverPublications. Zbl 0768.12001.

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Chapter 3

Algebraic extension

In abstract algebra, a field 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 coefficients in K. Field extensions that are not algebraic, i.e.which contain transcendental elements, are called transcendental.For example, the field extensionR/Q, that is the field of real numbers as an extension of the field of rational numbers,is transcendental, while the field extensionsC/R andQ(2)/Q are algebraic, whereC is the field of complex numbers.All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic.[1] Theconverse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraicnumbers is an infinite algebraic extension of the rational numbers.If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field:an algebraic extension of K which has finite degree over K. The converse is true as well, if K[a] is a field, then a isalgebraic over K. In the special case where K =Q is the field of rational numbers, Q[a] is an example of an algebraicnumber field.A field with no nontrivial algebraic extensions is called algebraically closed. An example is the field of complexnumbers. Every field 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 field.

3.1 Properties

The class of algebraic extensions forms a distinguished class of field 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 overfield 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 finitary results can be generalized using transfinite induction:

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

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

4

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3.2. GENERALIZATIONS 5

3.2 Generalizations

Main 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

{y N

p(y)}is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension.The Galois group of N overM can again be defined 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. 223ff, ISBN 978-0-201-55540-0, Zbl 0848.13001

McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of fields, 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

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Chapter 4

Degree of a field extension

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

4.1 Definition and notation

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

4.2 The multiplicativity formula for degrees

Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield 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 field 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 finite and infinite degree extensions. In the infinite case, the product is interpreted in thesense of products of cardinal numbers. In particular, this means that if M/K is finite, then both M/L and L/K arefinite.IfM/K is finite, then the formula imposes strong restrictions on the kinds of fields that can occur betweenM andK, viasimple arithmetical considerations. For example, if the degree [M:K] is a prime number p, then for any intermediatefield 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 fields (apart fromM and K themselves).

6

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4.2. THE MULTIPLICATIVITY FORMULA FOR DEGREES 7

4.2.1 Proof of the multiplicativity formula in the finite case

Suppose that K, L andM form a tower of fields as in the degree formula above, and that both d = [L:K] and e = [M:L]are finite. 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 findelements an in L such that

x =e

n=1

anwn = a1w1 + + aewe.

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

an =d

m=1

bm,num = b1,nu1 + + bd,nud.

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

x =e

n=1

(d

m=1

bm,num

)wn =

en=1

dm=1

bm,n(umwn),

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

0 =e

n=1

dm=1

bm,n(umwn)

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

0 =e

n=1

(d

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 =d

m=1

bm,num

for each n. Then, since the bm,n coefficients 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 infinite case

In 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 find that the products uwform a basis for M/K. These are indexed by the cartesian product A B, which by definition has cardinality equal tothe product of the cardinalities of A and B.

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8 CHAPTER 4. DEGREE OF A FIELD EXTENSION

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

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

The finite field GF(125) = GF(53) has degree 3 over its subfield 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 field extensionC(T)/C, whereC(T) is the field of rational functions overC, has infinite 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 field extension C(T2) also has infinite degree over C. However, if we view C(T2) as a subfield of C(T),then in fact [C(T):C(T2)] = 2. More generally, if X and Y are algebraic curves over a field K, and F : X Yis a surjective morphism between them of degree d, then the function fields K(X) and K(Y) are both of infinitedegree over K, but the degree [K(X):K(Y)] turns out to be equal to d.

4.4 Generalization

Given 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. Brieflydiscusses the infinite dimensional case.

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Chapter 5

Dual basis in a field extension

In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, byusing the field trace. This requires the property that the field 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 field, and hencein the cases where K is finite, 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 finite field, 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 = j1, otherwise

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

Tr() =m1i=0

pi

Using a dual basis can provide a way to easily communicate between devices that use different 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

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Chapter 6

Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with abase field and construct in some manner a larger field that contains the base field and satisfies 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 Definitions

Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under takinginverses in L. In other words, K is a field with respect to the field operations inherited from L. The larger field L isthen said to be an extension field of K. To simplify notation and terminology, one says that L / K (read as "L overK") is a field extension to signify that L is an extension field of K.If L is an extension of F which is in turn an extension ofK, then F is said to be an intermediate field (or intermediateextension or subextension) of the field extension L /K.Given a field extension L /K and a subset S of L, the smallest subfield of L which contains K and S is denoted byK(S)i.e. K(S) is the field 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 field extension of the form L = K(s) is called a simple extension and s is called a primitiveelement of the extension.Given a field extension L /K, the larger field 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 field 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 finite orinfinite the extension is called a finite extension or infinite 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 field extensions in situations where the small field is not actually contained in thelarger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an injective ringhomomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields donot possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields.Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.

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6.3. EXAMPLES 11

6.3 Examples

The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extensionfield of the field of rational numbersQ. Clearly then, C/Q is also a field extension. We have [C : R] = 2 because {1,i}is a basis, so the extension C/R is finite. This is a simple extension because C=R( i ). [R : Q] = c (the cardinality ofthe continuum), so this extension is infinite.The set Q(2) = {a + b2 | a, b Q} is an extension field 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 field of both Q(2) and Q, of degree 2 and 4 respectively. Finite extensions of Q are alsocalled algebraic number fields and are important in number theory.Another extension field of the rationals, quite different in flavor, is the field of p-adic numbersQp for a prime numberp.It is common to construct an extension field of a given field 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 field 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 field of any polynomial from K[X]. This is anextension field 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 finite field GF(pn) with pn elements; this is an extensionfield of the finite field GF(p) = Z/pZ with p elements.Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; theelements of K(X) are fractions of two polynomials over K, and indeed K(X) is the field of fractions of the polynomialring K[X]. This field of rational functions is an extension field of K. This extension is infinite.Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by C(M). It is anextension field of C, if we identify every complex number with the corresponding constant function defined on M.Given an algebraic varietyV over some fieldK, then the function field ofV, consisting of the rational functions definedon V and denoted by K(V), is an extension field of K.

6.4 Elementary properties

If L/K is a field 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 extensions

If 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 coefficients 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.

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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 coefficients 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 find 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 finite subextensions. In particular,every finite extension is algebraic. For example,

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

R/Q is transcendental, although not purely transcendental.

K(X)/K is purely transcendental.

A simple extension is finite 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 finite nor purely transcendental.

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

6.6 Normal, separable and Galois extensions

An 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 fieldof 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 field extension that is both normaland separable.A consequence of the primitive element theorem states that every finite separable extension has a primitive element(i.e. is simple).Given any field extensionL/K, we can consider its automorphismgroupAut(L/K), consisting of all field 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 field extension L/K, one is often interested in the intermediate fields F (subfields of L that contain K).The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediatefields: there is a bijection between the intermediate fields and the subgroups of the Galois group, described by thefundamental theorem of Galois theory.

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6.7. GENERALIZATIONS 13

6.7 Generalizations

Field 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 field, which are simple algebra (nonon-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the onlyfinite field 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 field is replaced by a commutative local ring.

6.8 Extension of scalars

Main article: Extension of scalars

Given a field extension, one can "extend scalars" on associated algebraic objects. For example, given a real vectorspace, one can produce a complex vector space via complexification. In addition to vector spaces, one can performextension of scalars for associative algebras defined over the field, such as polynomials or group algebras and theassociated group representations. Extension of scalars of polynomials is often used implicitly, by just considering thecoefficients as being elements of a larger field, 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 field theory

Tower of fields

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 field, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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Chapter 7

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently,E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significanceof 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 field, and G is a finite groupof automorphisms of E with fixed field F, then E/F is a Galois extension.

7.1 Characterization of Galois extensions

An important theorem of Emil Artin states that for a finite 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 field of a separable polynomial with coefficients 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 fixed field of a subgroup of Aut(E).

F is the fixed field of Aut(E/F).

There is a one-to-one correspondence between subfields of E/F and subgroups of Aut(E/F).

7.2 Examples

There are two basic ways to construct examples of Galois extensions.

Take any field E, any subgroup of Aut(E), and let F be the fixed field.

Take any field F, any separable polynomial in F[x], and let E be its splitting field.

Adjoining to the rational number field 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 first

14

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7.3. REFERENCES 15

of them is the splitting field of x2 2; the second has normal closure that includes the complex cube roots of unity,and so is not a splitting field. 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 fieldK is Galois overK if and only ifK is a perfect field.

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

7.4 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.

Bewersdorff, 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 field-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).

https://en.wikipedia.org/wiki/Normal_extensionhttps://en.wikipedia.org/wiki/Cube_roots_of_unityhttps://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theoryhttps://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theoryhttps://en.wikipedia.org/wiki/Algebraic_closurehttps://en.wikipedia.org/wiki/Perfect_fieldhttps://en.wikipedia.org/wiki/Galois_grouphttps://en.wikipedia.org/wiki/Emil_Artinhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-62342-4https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1616156https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-8218-3817-2https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=2251389https://en.wikipedia.org/wiki/Harold_Edwards_(mathematician)https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-90980-Xhttps://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=0743418https://en.wikipedia.org/wiki/Howard_G._Funkhouserhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2299273https://en.wikipedia.org/wiki/JSTORhttps://www.jstor.org/stable/2299273http://www.encyclopediaofmath.org/index.php?title=p/g043160https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Nathan_Jacobsonhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-7167-1480-9https://en.wikipedia.org/wiki/Cambridge_University_Presshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-80309-0https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-80309-0https://en.wikipedia.org/wiki/Grothendieckhttps://en.wikipedia.org/wiki/Groupoidshttps://en.wikipedia.org/wiki/Serge_Langhttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F978-1-4612-0853-2https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-94225-4https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1282723https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-43518-0https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=2043554https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1007%252F978-1-4612-0617-0https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-98541-7https://en.wikipedia.org/wiki/Special:BookSources/0-387-98541-7https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1645586https://en.wikipedia.org/wiki/Cambridge_University_Presshttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1017%252FCBO9780511471117https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-56280-5https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1405612https://en.wikipedia.org/wiki/Bartel_Leendert_van_der_Waerdenhttps://en.wikipedia.org/wiki/Florian_Pophttp://www.math.upenn.edu/~pop/Research/files-Res/Japan01.pdf

Chapter 8

Normal extension

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

8.1 Equivalent properties and examples

The 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, satisfies (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 finite extension of K that is separable (for example, this is automatically satisfied if K is finite 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 field for the polynomial.)

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

2) is

not a normal extension of Q since the irreducible polynomial x3 2 has one root in it (namely, 32 ), but not all of

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

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

field A of algebraic numbers is an algebraic closure of Q containing Q( 32) . On the other hand

Q( 32) = {a+ b 3

2 + c

34 A | a, b, c Q}

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

: Q( 32) A

a+ b 32 + c 3

4 7 a+ b 3

2 + c2 3

4

is an embedding ofQ( 32) inA whose restriction toQ is the identity. However, is not an automorphism ofQ( 3

2)

.For any prime p, the extension Q( p

2, p) is normal of degree p(p 1). It is a splitting field of xp 2. Here p

denotes any pth primitive root of unity. The field Q( 32, 3) is the normal closure (see below) of Q( 3

2) .

16

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8.2. OTHER PROPERTIES 17

8.2 Other properties

Let L be an extension of a field 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.

8.3 Normal closure

If K is a field 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 subfield 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 finite extension of K, then its normal closure is also a finite extension.

8.4 See also Galois extension

Normal basis

8.5 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

https://en.wikipedia.org/wiki/Compositumhttps://en.wikipedia.org/wiki/Up_to_isomorphismhttps://en.wikipedia.org/wiki/Galois_extensionhttps://en.wikipedia.org/wiki/Normal_basishttps://en.wikipedia.org/wiki/Serge_Langhttps://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematicshttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-387-95385-4https://en.wikipedia.org/wiki/Mathematical_Reviewshttp://www.ams.org/mathscinet-getitem?mr=1878556https://en.wikipedia.org/wiki/Nathan_Jacobsonhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-7167-1933-9https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1009787

Chapter 9

Separable extension

In the subfield of algebra named field theory, a separable extension is an algebraic field extension E F such thatfor every E , the minimal polynomial of over F is a separable polynomial (i.e., has distinct roots; see below forthe definition in this context).[1] Otherwise, the extension is called inseparable. There are other equivalent definitionsof 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 finite characteristic.More specifically, a finite degree field extension is Galois if and only if it is both normal and separable.[2] Sincealgebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstaclein most applications of Galois theory.[3][4] For instance, every algebraic (in particular, finite degree) extension of thefield 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 E \F , the minimal polynomial of over F is not a separable polynomial (i.e.,does not have distinct roots).[5] For a field F to possess a non-trivial purely inseparable extension, it must necessarilybe an infinite field of prime characteristic (i.e. specifically, imperfect), since any algebraic extension of a perfect fieldis necessarily separable.[3]

9.1 Informal discussion

An arbitrary polynomial f with coefficients in some field F is said to have distinct roots if and only if it has deg(f)roots in some extension field E F . For instance, the polynomial g(X)=X2+1 with real coefficients 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 coefficients 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 field 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 fieldof characteristic different 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 field 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 field of the coefficients, 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 coefficients may not have distinct roots, it is natural to ask atthis stage whether or not there exists an irreducible polynomial with rational or real coefficients 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 coefficients that does nothave distinct roots; in the language of field theory, every algebraic extension of Q or R is separable and hence bothof these fields are perfect.

18

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9.2. SEPARABLE AND INSEPARABLE POLYNOMIALS 19

9.2 Separable and inseparable polynomials

A 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 field in which its coefficients are considered to lie; forinstance, if g is an inseparable polynomial in F[X], and one considers a splitting field, 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 field 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 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 field 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 field has prime characteristic,for an irreducible polynomial to not have distinct roots its coefficients must lie in a field of prime characteristic.More generally, if an irreducible (non-zero) polynomial f in F[X] does not have distinct roots, not only must thecharacteristic of F be a (non-zero) prime number p, but also f(X)=g(Xp) for some irreducible polynomial g in F[X].[11]By repeated application of this property, it follows that in fact, f(X) = g(Xpn) for a non-negative integer n andsome separable irreducible polynomial g in F[X] (where F is assumed to have prime characteristic p).[12]

By the property noted in the above paragraph, if f is an irreducible (non-zero) polynomial with coefficients in thefield F of prime characteristic p, and does not have distinct roots, it is possible to write f(X)=g(Xp). Furthermore,if g(X) =

aiX

i , and if the Frobenius endomorphism of F is an automorphism, g may be written as g(X) =bpiX

i , and in particular, f(X) = g(Xp) =

bpiXpi = (

biX

i)p ; a contradiction of the irreducibility of f.Therefore, if F[X] possesses an inseparable irreducible (non-zero) polynomial, then the Frobenius endomorphism ofF cannot be an automorphism (where F is assumed to have prime characteristic p).[13]

If K is a finite field of prime characteristic p, and if X is an indeterminant, then the field of rational functions overK, K(X), is necessarily imperfect. Furthermore, the polynomial f(Y)=YpX is inseparable.[1] (To see this, note thatthere is some extension fieldE K(X) in which f has a root ; necessarily, p = X in E. Therefore, working overE, f(Y ) = Y p X = Y p p = (Y )p (the final equality in the sequence follows from freshmans dream),and f does not have distinct roots.) More generally, if F is any field of (non-zero) prime characteristic for which theFrobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.[14]

A field F is perfect if and only if all of its algebraic extensions are separable (in fact, all algebraic extensions of Fare separable if and only if all finite degree extensions of F are separable). By the argument outlined in the aboveparagraphs, it follows that F is perfect if and only if F has characteristic zero, or F has (non-zero) prime characteristicp and the Frobenius endomorphism of F is an automorphism.

9.3 Properties

IfE F is an algebraic field extension, and if , E are separable over F, then + and are separableover F. In particular, the set of all elements in E separable over F forms a field.[15]

If E L F is such that E L and L F are separable extensions, then E F is separable.[16]Conversely, if E F is a separable algebraic extension, and if L is any intermediate field, then E L andL F are separable extensions.[17]

If E F is a finite degree separable extension, then it has a primitive element; i.e., there exists E withE = F [] . This fact is also known as the primitive element theorem or Artins theorem on primitive elements.

https://en.wikipedia.org/wiki/Splitting_fieldhttps://en.wikipedia.org/wiki/Formal_derivativehttps://en.wikipedia.org/wiki/Frobenius_endomorphismhttps://en.wikipedia.org/wiki/Automorphismhttps://en.wikipedia.org/wiki/Imperfect_fieldhttps://en.wikipedia.org/wiki/Freshman%2527s_dreamhttps://en.wikipedia.org/wiki/Frobenius_endomorphismhttps://en.wikipedia.org/wiki/Perfect_fieldhttps://en.wikipedia.org/wiki/Frobenius_endomorphismhttps://en.wikipedia.org/wiki/Primitive_element_theorem

20 CHAPTER 9. SEPARABLE EXTENSION

9.4 Separable extensions within algebraic extensions

Separable extensions occur quite naturally within arbitrary algebraic field extensions. More specifically, if E Fis an algebraic extension and if S = { E| is separable over F} , then S is the unique intermediate field thatis separable over F and over which E is purely inseparable.[18] If E F is a finite degree extension, the degree [S: F] is referred to as the separable part of the degree of the extension E F (or the separable degree of E/F),and is often denoted by [E : F] or [E : F] .[19] The inseparable degree of E/F is the quotient of the degree bythe separable degree. When the characteristic of F is p > 0, it is a power of p.[20] Since the extension E F isseparable if and only if S = E , it follows that for separable extensions, [E : F]=[E : F] , and conversely. If E Fis not separable (i.e., inseparable), then [E : F] is necessarily a non-trivial divisor of [E : F], and the quotient isnecessarily a power of the characteristic of F.[19]

On the other hand, an arbitrary algebraic extensionE F may not possess an intermediate extension K that is purelyinseparable over F and over which E is separable (however, such an intermediate extension does exist when E Fis a finite degree normal extension (in this case, K can be the fixed field of the Galois group of E over F)). If suchan intermediate extension does exist, and if [E : F] is finite, then if S is defined as in the previous paragraph, [E :F] =[S : F]=[E : K].[21] One known proof of this result depends on the primitive element theorem, but there doesexist a proof of this result independent of the primitive element theorem (both proofs use the fact that if K F isa purely inseparable extension, and if f in F[X] is a separable irreducible polynomial, then f remains irreducible inK[X][22]). The equality above ([E : F] =[S : F]=[E : K]) may be used to prove that if E U F is such that [E :F] is finite, then [E : F] =[E : U] [U : F] .[23]

If F is any field, the separable closure Fsep of F is the field of all elements in an algebraic closure of F that areseparable over F. This is the maximal Galois extension of F. By definition, F is perfect if and only if its separable andalgebraic closures coincide (in particular, the notion of a separable closure is only interesting for imperfect fields).

9.5 The definition of separable non-algebraic extension fields

Although many important applications of the theory of separable extensions stem from the context of algebraic fieldextensions, there are important instances in mathematics where it is profitable to study (not necessarily algebraic)separable field extensions.Let F/k be a field extension and let p be the characteristic exponent of k .[24] For any field extension L of k, we writeFL = L k F (cf. Tensor product of fields.) Then F is said to be separable over k if the following equivalentconditions are met:

F p and k are linearly disjoint over kp

Fk1/p is reduced.

FL is reduced for all field extensions L of k.

(In other words, F is separable over k if F is a separable k-algebra.)A separating transcendence basis for F/k is an algebraically independent subset T of F such that F/k(T) is a finiteseparable extension. An extension E/k is separable if and only if every finitely generated subextension F/k of E/k hasa separating transcendence basis.[25]

Suppose there is some field extension L of k such that FL is a domain. Then F is separable over k if and only if thefield of fractions of FL is separable over L.An algebraic element of F is said to be separable over k if its minimal polynomial is separable. If F/k is an algebraicextension, then the following are equivalent.

F is separable over k.

F consists of elements that are separable over k.

Every subextension of F/k is separable.

Every finite subextension of F/k is separable.

https://en.wikipedia.org/wiki/Primitive_element_theoremhttps://en.wikipedia.org/wiki/Algebraic_closurehttps://en.wikipedia.org/wiki/Galois_extensionhttps://en.wikipedia.org/wiki/Characteristic_exponent_of_a_fieldhttps://en.wikipedia.org/wiki/Tensor_product_of_fieldshttps://en.wikipedia.org/wiki/Linearly_disjointhttps://en.wikipedia.org/wiki/Separable_algebrahttps://en.wikipedia.org/wiki/Algebraic_independence

9.6. DIFFERENTIAL CRITERIA 21

If F/k is finite extension, then the following are equivalent.

(i) F is separable over k.

(ii) F = k(a1, ..., ar) where a1, ..., ar are separable over k.

(iii) In (ii), one can take r = 1.

(iv) If K is an algebraic closure of k, then there are precisely [F : k] embeddings F into K which fix k.

(v) If K is any normal extension of k such that F embeds into K in at least one way, then there are precisely[F : k] embeddings F into K which fix k.

In the above, (iii) is known as the primitive element theorem.Fix the algebraic closure k , and denote by ks the set of all elements of k that are separable over k. ks is then separablealgebraic over k and any separable algebraic subextension of k is contained in ks ; it is called the separable closureof k (inside k ). k is then purely inseparable over ks . Put in another way, k is perfect if and only if k = ks .

9.6 Differential criteria

The separability can be studied with the aid of derivations and Khler differentials. Let F be a finitely generated fieldextension of a field k . Then

dimF Derk(F, F ) tr. degk F

where the equality holds if and only if F is separable over k.In particular, if F/k is an algebraic extension, then Derk(F, F ) = 0 if and only if F/k is separable.[26]

Let D1, ..., Dm be a basis of Derk(F, F ) and a1, ..., am F . Then F is separable algebraic over k(a1, ..., am) ifand only if the matrix Di(aj) is invertible. In particular, when m = tr. degk F , {a1, ..., am} above is called theseparating transcendence basis.

9.7 See also Purely inseparable extension

Perfect field

Primitive element theorem

Normal extension

Galois extension

Algebraic closure

9.8 Notes[1] Isaacs, p. 281

[2] Isaacs, Theorem 18.13, p. 282

[3] Isaacs, Theorem 18.11, p. 281

[4] Isaacs, p. 293

[5] Isaacs, p. 298

https://en.wikipedia.org/wiki/Primitive_element_theoremhttps://en.wikipedia.org/wiki/K%C3%A4hler_differentialhttps://en.wikipedia.org/wiki/Finitely_generated_field_extensionhttps://en.wikipedia.org/wiki/Finitely_generated_field_extensionhttps://en.wikipedia.org/wiki/Purely_inseparable_extensionhttps://en.wikipedia.org/wiki/Perfect_fieldhttps://en.wikipedia.org/wiki/Primitive_element_theoremhttps://en.wikipedia.org/wiki/Normal_extensionhttps://en.wikipedia.org/wiki/Galois_extensionhttps://en.wikipedia.org/wiki/Algebraic_closure

22 CHAPTER 9. SEPARABLE EXTENSION

[6] Isaacs, p. 280

[7] Isaacs, Lemma 18.10, p. 281

[8] Isaacs, Lemma 18.7, p. 280

[9] Isaacs, Theorem 19.4, p. 295

[10] Isaacs, Corollary 19.5, p. 296

[11] Isaacs, Corollary 19.6, p. 296

[12] Isaacs, Corollary 19.9, p. 298

[13] Isaacs, Theorem 19.7, p. 297

[14] Isaacs, p. 299

[15] Isaacs, Lemma 19.15, p. 300

[16] Isaacs, Corollary 19.17, p. 301

[17] Isaacs, Corollary 18.12, p. 281

[18] Isaacs, Theorem 19.14, p. 300

[19] Isaacs, p. 302

[20] Lang 2002, Corollary V.6.2

[21] Isaacs, Theorem 19.19, p. 302

[22] Isaacs, Lemma 19.20, p. 302

[23] Isaacs, Corollary 19.21, p. 303

[24] The characteristic exponent of k is 1 if k has characteristic zero; otherwise, it is the characteristic of k.

[25] Fried & Jarden (2008) p.38

[26] Fried & Jarden (2008) p.49

9.9 References Borel, A. Linear algebraic groups, 2nd ed.

P.M. Cohn (2003). Basic algebra

Fried,Michael D.; Jarden,Moshe (2008). Field arithmetic. Ergebnisse derMathematik und ihrer Grenzgebiete.3. Folge 11 (3rd ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.

I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. pp. 5559. ISBN 0-226-42451-0. Zbl 1001.16500.

M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese)

Silverman, Joseph (1993). The Arithmetic of Elliptic Curves. Springer. ISBN 0-387-96203-4.

9.10 External links Hazewinkel, Michiel, ed. (2001), separable extension of a field k, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

https://en.wikipedia.org/wiki/Separable_extension#CITEREFLang2002https://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-77269-9https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:1145.12001https://en.wikipedia.org/wiki/Martin_Isaacshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-534-19002-2https://en.wikipedia.org/wiki/Special:BookSources/0-534-19002-2https://en.wikipedia.org/wiki/Irving_Kaplanskyhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-226-42451-0https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:1001.16500https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-96203-4http://www.encyclopediaofmath.org/index.php?title=s/s084470https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4

Chapter 10

Simple extension

In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simpleextensions are well understood and can be completely classified.The primitive element theorem provides a characterization of the finite simple extensions.

10.1 Definition

A field extension L/K is called a simple extension if there exists an element in L with

L = K().

The element is called a primitive element, or generating element, for the extension; we also say that L is generatedover K by .Every finite field is a simple extension of the prime field of the same characteristic. More precisely, if p is a primenumber and q = pd the field Fq of q elements is a simple extension of degree d of Fp. T