ring homomorphism

Ring homomorphismFrom 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

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 Ring homomorphism 189.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 The category of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.3.1 Endomorphisms, isomorphisms, and automorphisms . . . . . . . . . . . . . . . . . . . . . 209.3.2 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

10 Separable extension 2110.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.5 The denition of separable non-algebraic extension elds . . . . . . . . . . . . . . . . . . . . . . 2310.6 Dierential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

CONTENTS iii

11 Simple extension 2611.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.2 Structure of simple extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

12 Tower of elds 2812.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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 of

complex 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 no

non-trivial elds between them. The eld extension Q(2, 3), obtained by adjoining 2 and 3 to the eld Q of rational numbers, has degree

4, 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. Proof

of the multiplicativity formula. page 465, Jacobson, N. (1989). Basic Algebra II. W. H. Freeman and Company. ISBN 0-7167-1933-9. Briey

discusses 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 (see

transcendental 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 third

ed.), 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

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.

7.1 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).

7.2 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

14

7.3. REFERENCES 15

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.

7.3 References[1] See the article Galois group for denitions 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. Bewersdor, Jrg (2006). Galois theory for beginners. Student Mathematical Library 35. Translated from

the 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 8

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.

8.1 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) .

16

8.2. OTHER PROPERTIES 17

8.2 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.

8.3 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.

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

Chapter 9

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.

9.1 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 group

homomorphism 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 R

arises 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.

18

9.2. EXAMPLES 19

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 initial

object 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.

9.2 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).

9.3 The category of ringsMain article: Category of rings

20 CHAPTER 9. RING HOMOMORPHISM

9.3.1 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.

9.3.2 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.

9.4 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.

9.5 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.

9.6 See also change of rings

Chapter 10

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]

10.1 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.

21

22 CHAPTER 10. SEPARABLE EXTENSION

10.2 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 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 coecients in theeld F of prime characteristic p, and does not have distinct roots, it is possible to write f(X)=g(Xp). Furthermore,if g(X) = P aiXi , and if the Frobenius endomorphism of F is an automorphism, g may be written as g(X) =P

bpiXi , and in particular, f(X) = g(Xp) = P bpiXpi = (P biXi)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 nite eld of prime characteristic p, and if X is an indeterminant, then the eld 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 eldE 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 nal equality in the sequence follows from freshmans dream),and f does not have distinct roots.) More generally, if F is any eld of (non-zero) prime characteristic for which theFrobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.[14]

A eld 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 nite 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.

10.3 Properties

IfE F is an algebraic eld extension, and if ; 2 E are separable over F, then + and are separableover F. In particular, the set of all elements in E separable over F forms a eld.[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 eld, then E L andL F are separable extensions.[17]

If E F is a nite degree separable extension, then it has a primitive element; i.e., there exists 2 E withE = F [] . This fact is also known as the primitive element theorem or Artins theorem on primitive elements.

10.4. SEPARABLE EXTENSIONS WITHIN ALGEBRAIC EXTENSIONS 23

10.4 Separable extensions within algebraic extensionsSeparable extensions occur quite naturally within arbitrary algebraic eld extensions. More specically, if E Fis an algebraic extension and if S = f 2 Ej is separable over Fg , then S is the unique intermediate eld thatis separable over F and over which E is purely inseparable.[18] If E F is a nite 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 nite degree normal extension (in this case, K can be the xed eld of the Galois group of E over F)). If suchan intermediate extension does exist, and if [E : F] is nite, then if S is dened 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 nite, then [E : F]=[E : U][U : F].[23]

If F is any eld, the separable closure Fsep of F is the eld of all elements in an algebraic closure of F that areseparable over F. This is the maximal Galois extension of F. By denition, 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 elds).

10.5 The denition of separable non-algebraic extension eldsAlthough many important applications of the theory of separable extensions stem from the context of algebraic eldextensions, there are important instances in mathematics where it is protable to study (not necessarily algebraic)separable eld extensions.Let F/k be a eld extension and let p be the characteristic exponent of k .[24] For any eld extension L of k, we writeFL = L k F (cf. Tensor product of elds.) 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 eld 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 niteseparable extension. An extension E/k is separable if and only if every nitely generated subextension F/k of E/k hasa separating transcendence basis.[25]

Suppose there is some eld extension L of k such that FL is a domain. Then F is separable over k if and only if theeld 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 nite subextension of F/k is separable.

24 CHAPTER 10. SEPARABLE EXTENSION

If F/k is nite 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 x 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 x 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 .

10.6 Dierential criteriaThe separability can be studied with the aid of derivations and Khler dierentials. Let F be a nitely generated eldextension of a eld 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 2 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 , fa1; :::; amg above is called theseparating transcendence basis.

10.7 See also Purely inseparable extension Perfect eld Primitive element theorem Normal extension Galois extension Algebraic closure

10.8 Notes[1] Isaacs, p. 281

[2] Isaacs, Theorem 18.13, p. 282


[4] Isaacs, p. 293

[5] Isaacs, p. 298

10.9. REFERENCES 25

[6] Isaacs, p. 280

[7] Isaacs, Lemma 18.10, p. 281

[8] Isaacs, Lemma 18.7, p. 280


[10] Isaacs, Corollary 19.5, p. 296




[14] Isaacs, p. 299

[15] Isaacs, Lemma 19.15, p. 300




[19] Isaacs, p. 302

[20] Lang 2002, Corollary V.6.2


[22] Isaacs, Lemma 19.20, p. 302


[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

10.9 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 eld theory: new edition, Shokado. (Japanese) Silverman, Joseph (1993). The Arithmetic of Elliptic Curves. Springer. ISBN 0-387-96203-4.

10.10 External links Hazewinkel, Michiel, ed. (2001), separable extension of a eld k, Encyclopedia of Mathematics, Springer,

ISBN 978-1-55608-010-4

Chapter 11

Simple extension

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

11.1 DenitionA eld 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 nite eld is a simple extension of the prime eld of the same characteristic. More precisely, if p is a primenumber and q = pd the eld Fq of q elements is a simple extension of degree d of Fp: This means that it is generatedby an element which is a root of an irreducible polynomial of degree d. However, in this case, is normally notreferred to as a primitive element.In fact, a primitive element of a nite eld is usually dened as a generator of the elds multiplicative group. Moreprecisely, by little Fermat theorem, the nonzero elements of Fq (i.e. its multiplicative group) are the roots of theequation

xq1 1 = 0;

that is the (q1)-th roots of unity. Therefore, in this context, a primitive element is a primitive (q1)-th root ofunity, that is a generator of the multiplicative group of the nonzero elements of the eld. Clearly, a group primitiveelement is a eld primitive element, but the contrary is false.Thus the general denition requires that every element of the eld may be expressed as a polynomial in the generator,while, in the realm of nite elds, every nonzero element of the eld is a pure power of the primitive element. Todistinguish these meanings one may use eld primitive element of L over K for the general notion, and groupprimitive element for the nite eld notion.[1]

11.2 Structure of simple extensionsIf L is a simple extension of K generated by , it is the only eld contained in L which contains both K and .This means that every element of L can be obtained from the elements of K and by nitely many eld operations(addition, subtraction, multiplication and division).

26

11.3. EXAMPLES 27

Let us consider the polynomial ring K[X]. One of its main properties is that there exists a unique ring homomorphism

' : K[X] ! Lp(X) 7! p() :

Two cases may occur.If ' is injective, it may be extended to the eld of fractions K(X) of K[X]. As we have supposed that L is generatedby , this implies that ' is an isomorphism from K(X) onto L. This implies that every element of L is equal to anirreducible fraction of polynomials in , and that two such irreducible fractions are equal if and only if one may passfrom one to the other by multiplying the numerator and the denominator by the same non zero element of K.If ' is not injective, let p(X) be a generator of its kernel, which is thus the minimal polynomial of . The imageof ' is a subring of L, and thus an integral domain. This implies that p is an irreducible polynomial, and thus thatthe quotient ring K[X]/hpi is a eld. As L is generated by , ' is surjective, and ' induces an isomorphism fromK[X]/hpi onto L. This implies that every element of L is equal to a unique polynomial in , of degree lower thanthe degree of the extension.

11.3 Examples C:R (generated by i) Q(2):Q (generated by 2), more generally any number eld (i.e., a nite extension ofQ) is a simple extensionQ() for some . For example, Q(

p3;p7) is generated by

p3 +

p7 .

F(X):F (generated by X).

11.4 References[1] (Roman 1995)

Roman, Steven (1995). Field Theory. Graduate Texts inMathematics 158. NewYork: Springer-Verlag. ISBN0-387-94408-7. Zbl 0816.12001.

Chapter 12

Tower of elds

In mathematics, a tower of elds is a sequence of eld extensions

F0 F1 ... Fn ...

The name comes from such sequences often being written in the form

...jF2jF1jF0:

A tower of elds may be nite or innite.

12.1 Examples Q R C is a nite tower with rational, real and complex numbers. The sequence obtained by letting F0 be the rational numbers Q, and letting

Fn+1 = Fn

21/2

n

(i.e. Fn is obtained from Fn by adjoining a 2n th root of 2) is an innite tower.

If p is a prime number the p th cyclotomic tower ofQ is obtained by letting F0 =Q and Fn be the eld obtainedby adjoining to Q the pn th roots of unity. This tower is of fundamental importance in Iwasawa theory.

The GolodShafarevich theorem shows that there are innite towers obtained by iterating the Hilbert class eldconstruction to a number eld.

12.2 References Section 4.1.4 of Escoer, Jean-Pierre (2001), Galois theory, Graduate Texts in Mathematics 204, Springer-

Verlag, ISBN 978-0-387-98765-1

28

12.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 29

12.3 Text and image sources, contributors, and licenses12.3.1 Text

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Field extension Source: https://en.wikipedia.org/wiki/Field_extension?oldid=655247312 Contributors: AxelBoldt, Zundark, Edward,TakuyaMurata, Daran, Naddy, Lowellian, MathMartin, Giftlite, Fropu, Mazi, El C, Bookofjude, EmilJ, Oleg Alexandrov, Marudub-shinki, Graham87, FlaBot, YurikBot, Dmharvey, Mathaxiom~enwiki, Grubber, KnightRider~enwiki, SmackBot, Nbarth, Foxjwill, Ewjw,Jim.belk, Madmath789, CRGreathouse, Thijs!bot, RobHar, Escarbot, JAnDbot, Magioladitis, David Eppstein, Cpiral, Policron, STBotD,PerezTerron, TXiKiBoT, Don4of4, AlleborgoBot, Cwkmail, JackSchmidt, Yasmar, Ideal gas equation, Mpd1989, Alexbot, He7d3r, Ad-dbot, PV=nRT, Ptbotgourou, Calle, Xqbot, Point-set topologist, Uuo, Sawomir Biay, Vanished user jtji34toksdcknqrjn54yoimascj,Cenkner, YFdyh-bot and Anonymous: 46

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Ring homomorphism Source: https://en.wikipedia.org/wiki/Ring_homomorphism?oldid=664636173 Contributors: AxelBoldt, BryanDerksen, TakuyaMurata, Charles Matthews, Mtheory~enwiki, Robbot, Mattblack82, Altenmann, Giftlite, Lethe, Fropu, Ssd, Touriste,Mdd, Oleg Alexandrov, Isnow, Mathbot, YurikBot, Zarel, David Molenaer, Teply, Vina-iwbot~enwiki, CmdrObot, WinBot, Magioladi-tis, Albmont, JoergenB, Don4of4, Niv.sarig, Alexbot, He7d3r, Lunarmono, Ozob, PV=nRT, Legobot, Yobot, Xqbot, WissensDrster,Erik9bot, FrescoBot, Ebony Jackson, MastiBot, De Charlus, Quondum, ChuispastonBot, ClueBot NG, KEmilG, Magus145, Atari11,GeoreyT2000, JDude13 and Anonymous: 35

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30 CHAPTER 12. TOWER OF FIELDS

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Abelian extensionReferences

Algebraic closureExamplesExistence of an algebraic closure and splitting fieldsSeparable closureSee alsoReferences

Algebraic extensionPropertiesGeneralizationsSee also Notes References

Degree of a field extensionDefinition and notation The multiplicativity formula for degrees Proof of the multiplicativity formula in the finite case Proof of the formula in the infinite case

Examples Generalization References

Dual basis in a field extensionField extensionDefinitions Caveats Examples Elementary properties Algebraic and transcendental elements and extensions Normal, separable and Galois extensions Generalizations Extension of scalars See also Notes ReferencesExternal links

Galois extensionCharacterization of Galois extensionsExamplesReferences See also

Normal extensionEquivalent properties and examples Other properties Normal closureSee also References

Ring homomorphismProperties Examples The category of ringsEndomorphisms, isomorphisms, and automorphismsMonomorphisms and epimorphisms

Notes References See also

Separable extensionInformal discussionSeparable and inseparable polynomialsPropertiesSeparable extensions within algebraic extensionsThe definition of separable non-algebraic extension fields Differential criteria See alsoNotesReferencesExternal links

Simple extensionDefinition Structure of simple extensions Examples References

Tower of fieldsExamplesReferencesText and image sources, contributors, and licensesTextImagesContent license

ring homomorphism

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