A book of abstract algebra by charles c pinter (2nd ed)

A BOOK OF ABSTRACT ALGEBRA Charles C Pinter Professor of Mathematics Bucknell University McGraw-Hili Book Company New York St Louis San Francisco Auckland Bogota Hamburg Johannesburg London Madrid Mexico Montrcal New Delhi Panama Paris SlIo Paulo Singapore Sydney Tokyo Toronto This book was set in Times Roman by Santype-Byrd The editors were John J Corrigan and James S Arnar; the production supervisor was Leroy A Young The drawings were done by VIP Graphics The cover was designed by Scott Che1ius R R Donnelley & Sons Company was printer and binder A BOOK OF ABSTRACf ALGEBRA Copyright 10 1982 by McGraw-Hili, Inc All rights reserved Printed in the United States of America Except as pennitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written pennission of the publisher 1234567890 DODO 8987654321 ISBN 0-07-050130-0 Ubrllry of Congress Clltaloging in Publication Dlltll Pinter, Charles C, date A book of abstract algebra Includes index \ Algebra, Abstract QA162.P56 512'.02 ISB N 0-07-050130-0 I Title 81-8420 AACR:! To my colleagues in Brazil, especially Newton da Costa, Ayda Arruda, and Elias Alves, as well as to others In appreciation of their loyal and valued friend ship CONTENTS' Preface xiii Chapter Why Abstract Algebra? History of Algebra New Algebras Algebraic Structures Axioms and Axiomatic Algebra Abstractio n in Algebra Cbapter Operations 20 Operations o n a Set Properties of Operations Chapter The Definition of Groups 26 Groups Examples of Infinite and Finite Groups Eumples of Abelian and Nonabelian Groups Group Tables Chapter Elementary Properties of Groups 36 Uniqueness of Identity and Inverses Properties of Inverses Direct Product of Groups Subgroups 45 Definition of Subgroup Generators and Defining Relations Cayley Diagrams Center of Q Group • Italic beadings indicate topics discussed in the exercise sections vii yiii CONTENTS Chapter Functions 54 Injective,.Surjective, Bijective Function Composite and Inverse of Functions Chapter Groups of Permutations Symmetric Groups 64 Dihedral Groups Chapter Permutations of a Finite Set 75 Decomposition of Permutations into Cycles Transpositions Even and Odd Permutations Alternating Groups Chapter Isomorphism 86 The Concept of Isomorphism in Mathematics Isomorphic and Nonisomorphic Groups Cayley's Theorem Group Automorphistm Chapter 10 Order of Group Elements 100 Powers/ Multiples of Group Elements Laws of Exponents Properties of the Order of Group Elements Chapter 11 Cyclic Groups 109 Finite and Infinite Cyclic Groups Isomorphism of Cyclic Groups Subgroups of Cyclic Groups Chapter 12 Partitions and Equivalence Relations 117 Chapter 13 Counting Cosets 123 Lagrange's Theorem and Elementary Consequences Number a/Conjugate Elements Group Acting on a Set Survey a/Group$ a/Order.:S 10 Chapter 14 Homomorphisms Elementary Properties of Homomorphism Subgroups Kernel and Range Inner Direct Products Conjugate Subgroup$ 132 Normal CONTENTS ix Chapter 15 Quotient Groups Quotient Group Construction Applications The Class Equation 143 Exam,ples and Induction on the Order of a Group Chapter 16 The Fundamental Homomorphism Theorem 153 Fundamental Homomorphism Theorem and Some Consequences The Isomorphism Theorems The Corre$pomience Theorem Cauchy'$ Theorem Sylow Subgroups Sylow's Theorem Decomposition Theorem for Finite Abelian Groups Chapter 17 Rings: Definitions and Elementary Properties 165 Commutative Rings Unity Invertibles and Zero-Divisors Integral Domain Field Chapter 18 Ideals and Homomorphisms 178 Chapter 19 Quotient Rings 187 Construction of Quotient Rings Examples Fundamental Homomorphism Theorem and Some Consequences Properties of Prime and Maximal Ideals Chapter 20 Integral Domains 197 Characteristic of an Integral Domain Properties of the Characteristic Finite Fields Construction of the Field of Quotients Chapter 21 The Integers 205 Ordered Integral Domains Well-ordering Characterization of Z Up to Isomorphism Mathematical Induction Division Algorithm Chapter 22 Factoring into Primes Ideals of Z Properties of the GCn Relatively Prime Integers Primes Euclid's Lemma Unique Factorization 215 x CONTENTS Chapter 23 Elements of Number Theory 224 Properties of Congruence Theorems of Fermat and Euler Solutions of Linear Congruences Chinese Remainder Theorem Wilson's Theorem and Consequences Quadratic Residues The Legendre Symbol Primitive Roots Chapter 24 Rings of Polynomials 239 Motivation and Definitions Domain of Polynomials over a Field Division Algorithm Polynomials in Several Voriables Chapter 25 Field of Polynomial Quoliem s Factoring Polynomials 251 Ideals of F[x) Properties of the GCD Polynomials Unique factorization Irreducible Euclidean Algorithm Chapter 26 Substitution in Polynomials 258 Roots and Factors Polynomial Functions Polynomials over O Eisenstein's Irreducibility Criterion Polynomials over the Reals Polynomial Interpolation Chapter 27 Extensions of Fields 270 Algebraic and Transcendental Elements The Minimum Polynomial Basic Theorem on Field Extensions Chapter 28 Vector Spaces 282 Elementary Properties of Vector Spaces Linear Independence Basis Dimension Linear Transformations Chapter 29 Degrees of Field Extensions Simple and Iterated Extensions Extension Field of Algebraic Elements C/osll1'e 292 Degree of an Iterated Algebraic Numbers Algebraic CONTENTS Chapter 30 Ruler and Compass xi 301 Constructible Points and Numbers Constructions Impossible Conslructible Angles and Polygons Chapler 31 Galois Theory: Preamble Multiple Roots Root Field Isomorphism ROOfs of Unity 311 Extension of a Field Separable Polynomials Normal ExtensiollS Chapler 32 Galois Theory: The Hearl of The Matter 323 Field Automorphisms The Galois Group The Galois Correspondence Fundamental Theorem of Galois Theory Computing Galois Groups Chapler 33 Solving Equations by Radicals Radical Extensions Abelian Extensions Groups Insolvability of the Quintic Index 335 Solvable 347 SOLVIt-;"G EQUATIONS BY RADICALS 337 Theorem Any finite group of nonzero elements in a field is a cyclic group (The operation in the group is thefield's multiplication.) If F* denotes the set of nonzero elements of F suppose that Go;:: F*, and that G, with the field's " multiply" operation, is a group of n elements We will compare G with Z ~ and show that G, like Zn, has an element of order n and is therefore cyclic For every positive integer k which divides n, the equation Xk = has Qt most k solutions in F; thus, G contains k or fewer elements of order k On the other hand, in Zn there are exactly k elements of order k, namely Now, every element of G (as well as every element of Zn) has a welldefined order, which is a divisor of n Imagine the elements of both groups to be partitioned into classes according to their order, and compare the classes in G with the corresponding classes in Zn For each k, G has as many or fewer elements of order k than Zn does So if G had no elements of order n (while Zn does have one), this would mean that G has fewer elements than Zn, which is false Thus, G must have an element of order n, and therefore G is cyclic The nth roots of unity (which are contained in F or a suitable extension of F) obviously form a group with respect to multiplication By Theorem 1, it is a cyclic group Any generator of this group is called a primitive nth root of unity Thus, if OJ is a primitive nth root of unity, the set of all the nth roots of unity is If £0 is a primitive nth root of unity, F(£O) is an abelian extension of Fin the sense that h = II for any two F-fixing automorphisms and h of F(£O) Indeed, any automorphism must obviously send nth roots of unity to nth roots of unity So if g(£O) = £0' and h(£O) = £0$, then h(£O) = g{£O~) = £0", and analogously, II g(£O) = £0" Thus, h(£O) = h g(w) Since and" fix F, and every element of F(w) is a linear combination of powers of w with coefficients in F, g o " = h o g Now, let F contain a primitive nth root of unity, and therefore all the nth roots of unity Suppose a E F, and a has an nth root b in F It follows, then, that all the nth roots of a are in F,lor they are b, bw, bw , •••• b£On - ! Indeed if c is any other nth root of a, then clearly c/ b is an nth root of say 338 CHAPTER THIRTY -THREE ro', hence c :; bro' We may infer from the above that if F contains a primi· tive nth root of unity, and b is an nth root of a, then F(b) is the root field of x" - a over F In particular, F(b) is an abelian extension of F Indeed, any F -fixing automorphism of F(b) must send nth roots of a to nth roots of a: for if c is any nth root of a and is an F-fixing automorphism, then g(et :; g(e"} :; g(a) :; a, hence g(c) is an nth root of a So if g(b) = bro' and h(b) ~ bw', then g o h(b) = g(bmS) = bro'ro s = bof+ s and h g(b} = h(bro') = bafro' bm'+1 = hence h(b) = h g(b) Since and II fix F, and every element in F(b) is a linear combination of po wers of b with coefficients in F, it follows that g o lt=h o g If a(x) is in F[x], remember that a(x) is solvable by radicals just as long as there exists some radical extension of F containing the roots of a(x) [Any radical extension of F containing the roots of a(x) will do.] Thus, we may as well assume that any radical extension used here begins by adjoining to F the appropriate roots of unity; henceforth we will make this assumption Thus, if K = F(clt , cn) is a radical extension of F then F 10 ~ F(cd £; F(clt Cl) - ~ II 12 £; • ~ F(cl , cll) \ V ' I",=K is a sequence of simple abelian extensions (The extensions are all abelian by the commen ts in the preceding three paragraphs.) If G denotes the Galois group of Kover F, each of these fields I t has a fixer which is a subgroup of G These fixers form a sequence K* c 1:_ £; •• £; It £; F* For each k by Theorem o n page 330, I: is a normal subgroup of 1: + , and 1:+t!J: ~ Gaf(lk+I: I t} which is abelian because I H I is an abelian extension of I • The following definition was invented precisely to account for this sit uation : A group G is called solvable if il has a sequence of subgroups {e} = HI£; £; H", = G such that Jor each k G is a normal subgroup ofG k + and G~ + I/Gk is abelian Ho c:; SOLVING EQUATIONS BV RADICALS 339 We have shown that if K is a radical extension of F, then Gal(K : F) is a sO/V{lble group We wish to go further and prove that if a(x) is any polynomial which is solvable by radicals, its Galois group is solvable To so, we must first prove that any homomorphic image of a solvable group is solvable A key ingredient of our proof is the following simple fact, which was explained on page 148: GIH is abelian iff H contains all the products xyx - 1y - l, for all x and y in G (The products xyx-1y - are called "commutators" of G.) Theorem Any homomorphic image of a solvable group is a ~olvable group Let G be a solvable group, with a sequence of subgroups {e}£;H1£;"·£;Hm=G as specified in the definition Let f : G _ X be a homomorphism from G onto a group X Thenf(H o),f(Hd, f(H"') are subgroups of X, and clearly {e} £; f(Ho) £;f(H d£;' " £; f(H",) = X For each i we have the following : if f(a) E f( H 1) and f(x) E f(H j + I ), then a E H i and x E Hi+J, hence xax- I E H i and therefore f(x)f(a)f(x) - I E f(H/) So f(R I ) is a normal subgroup of f(H i +I)' Finally, since H ;+ d H I is abelian, every commutator xyx-1y-l (for all x and y in H i+ d is in Hi, hence every J(x)J(y)J(x) - 'J(y)-' is inJ(H,) Thus,f(H,,,)/J(H,) is abelian Now we can prove the main result of this chapter: Theorem Let a(x) be a polynomial over a field F If a(x) is solvable by radicals, ils Galois group is a solvable group By definition, if K is the root field of a(x), there is an extension by radicals F(cJ, , c,,) such that F £; K £; F(c., ,c,,) It follows by Theorem on page 330 that Gul(F(e" , c.) : F)jGal(F(e" , c.): K )" Gal(K : F), hence by Theorem on page 330, Gal(K : F) is a homomorphic image of Gal(F(ch , c,,): F) which we know to be solvable Thus, by Theorem Gal(K : F) is solvable Actually, the converse of Theorem is true also All we need to show is that, if K is an extension of F whose Galois group over F is solvable, then K may be further extended to a radical extension of F The details are not too difficult and are assigned as Exercise E at the end of this chapter Theorem together with its converse say that a polynomial a(x) is solvable by radicals iff its Galois group is solvable We bring this chapter to a close by showing that there exist groups 340 CHAPTER THIRTY-THREE which are not solvable, and there exist polynomials having such groups as their Galois group In other words, there are unsolvable polynomials First, here is an unsolvable group: Theorem The symmetric group S5 is not a solvable group Suppose S5 has a sequence of subgroups {e} = Ho S;; HI s;; ~ H", = Ss as in the definition of solvable group Consider the subset of Ss containing all the cycles (ijk) of length We will show that if Hi contains all the cycles of length 3, so does the next smaller group HI _ I It would follow in m steps that Ho = {e} contains all the cycles of length 3, which is absurd So let Hi contain all the cycles of length rn S5 Remember that if a and p are in Hi , then their commutator apa- i p- is in H _ I • But any cycle (ijk) is equal to the commutator (iljXjkm){ilj) - I (jkm) - I = (ilj){jkm){jIiXmkJ) = (ijk) hence every (ijk) is in Hi _ I, as claimed Before drawing our argument towaTd a close, we need to know one more fact about groups; it is contained in the following classical result of group theory: Cauchy's Iheorem Let G be a finite group oj n elements number which divides n, then G has an element oj order p IJ p is any prime For example, if G is a group of 30 elements, it has elements of orders 2, 3, and To give our proof a trimmer appearance, we will prove Cauchy's theorem specifically for p = (the only case we will use here, anyway) However, the same argument works for any value ofp Consider all possible 5-tuples (a, b, c, d, k) of elements of G whose product abcdk = e How.many distinct 5-tuples of this kind are there? Well, if we select a, b, c, and d at random, there is a unique k = d- IC - Ib - la - in G making abcdk = e Thus, there are n4 such 5-tuples Call two 5-tuples equivalent if one is merely a cyclic permutation of the other Thus, (a, b, c, d, k) is equivalent to exactly five distinct 5-tuples, namely (a, b, c, d, k), (b, c, d, k a), (c, d, k, a, b), (d, k, a, b, c) and (k, a, b, c, d) The only exception occurs when a 5-tuple is of the form (a, a, a, a, a) with all its components equal; it is equivalent only to itself Thus, the equivalence class of any 5.tuple of the fonn (a, a, a, a a) has a single member, while all the other equivalence classes have five members SOLVING EQUATIONS BY RADICALS 34 Are there any equivalence classes, other than {(e, e, e, e, e)}, wit h a single member? If nOl then SI(n - 1) (for there are n4 S-tuples under consideration, less (e, e, e, e, e», hence n4 == (mod 5) But we are assuming that I n, hence n4 == (mod 5), which is a contradiction This contradiction shows that there must be a 5-tuple (a, a, a, a, a) #= (e, e, e, e, e) such that aaaaa = a = e Thus, there is an element a E G of order We wi ll now exhibit a polynomial in Q [x] having S~ as its Galois group (remember that S~ is nOl a solvable group) Let a(x) = x - Sx - By Eisenstein's criterion, a(x + 2) is irreducible over 0, hence a(x) also is irreducible over O By elementary calcu lus, a(x) has a single maximum at ( - 1,2), a single minimum at (I, - 6), and a single point of inflection at (0, - 2) Thus (see figu re), its graph intersects the x axis exactly three times This means that a(x) has three real roots, '1, '2, and,), and therefore two complex roots,'4 and ' 5, which must be complex conjugates of each other Let K denote the root field of a(x) over Q, and G the Galois group of a(x) As we have already noted, every element of G may be identified with a y=x' -5x_2 (- 1, 2) ~2 ~I ~I ~2 (0, _ 2) ~3 ~4 ~S ~6 (I, -6) 342 CHAPTER THIRTY-THREE permutation of the roots'l> '2, '3, '4, r5 of a(x), so G may be viewed as a subgroup of S5 We will show that G is all of S5 Now, [O(rd: 0] = because'l is a root of an irreducible polynomial of degree over O Since [K : ] = [K : O(rl)][Q('I): ] , it follows that is a factor of [K : OJ Then, by Cauchy's theorem, G contains an element of order This element must be a cycle of length 5: for by Chapter 8, every other element of 55 is a product of two or more disjoint cycles of length < and such products cannot have order (Try the typical cases.) Thus, G contains a cycle of length Furthermore G contains a transposition because com plex conjugation a + hi - a - hi is obviously a O -fixing automorphism of K; it interchanges '1.'2 and,) fixed the complex rootS'4 and'5 while leaving Any subgroup G