Contemporary Abstract Algebra SEVENTH EDITION Joseph A Gallian University of Minnesota Duluth Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Contemporary Abstract Algebra, Seventh Edition Joseph A Gallian VP/Editor-in-Chief: Michelle Julet Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor Associate Editor: Daniel Seibert Editorial Assistant: Shaylin Walsh © 2010, 2006 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher Managing Media Editor: Sam Subity Senior Content Manager: Maren Kunert Executive Marketing Manager: Joe Rogove Marketing Specialist: Ashley Pickering Marketing Communications Manager: Mary Anne Payumo Senior Content Project Manager, Editorial Production: Tamela Ambush For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be e-mailed to permissionrequest@cengage.com Senior Manufacturing Coordinator: Diane Gibbons Senior Rights Acquisition Account Manager: Katie Huha Library of Congress Control Number: 2008940386 Production Service: Matrix Productions Inc Student Edition: ISBN-13: 978-0-547-16509-7 Text Designer: Ellen Pettengell Design ISBN-10: 0-547-16509-9 Photo Researcher: Lisa Jelly Smith Cover Designer: Elise Vandergriff Cover Image: © Anne M Burns Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA Compositor: Pre-PressPMG Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at www.cengage.com/international Cengage Learning products are represented in Canada by Nelson Education, Ltd To learn more about Brooks/Cole , visit www.cengage.com/brookscole Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 12 11 10 09 08 Contents Preface xi PART Integers and Equivalence Relations Preliminaries Properties of Integers | Modular Arithmetic | Mathematical Induction 12 | Equivalence Relations 15 | Functions (Mappings) 18 Exercises 21 Computer Exercises 25 PART Groups 27 Introduction to Groups 29 Symmetries of a Square 29 | The Dihedral Groups 32 Exercises 35 Biography of Niels Abel 39 Groups 40 Definition and Examples of Groups 40 | Elementary Properties of Groups 48 | Historical Note 51 Exercises 52 Computer Exercises 55 Finite Groups; Subgroups 57 Terminology and Notation 57 | Subgroup Tests 58 | Examples of Subgroups 61 Exercises 64 Computer Exercises 70 iii iv Contents Cyclic Groups 72 Properties of Cyclic Groups 72 | Classification of Subgroups of Cyclic Groups 77 Exercises 81 Computer Exercises 86 Biography of J J Sylvester 89 Supplementary Exercises for Chapters 1–4 91 Permutation Groups 95 Definition and Notation 95 | Cycle Notation 98 | Properties of Permutations 100 | A Check Digit Scheme Based on D5 110 Exercises 113 Computer Exercises 118 Biography of Augustin Cauchy 121 Isomorphisms 122 Motivation 122 | Definition and Examples 122 | Cayley’s Theorem 126 | Properties of Isomorphisms 128 | Automorphisms 129 Exercises 133 Computer Exercise 136 Biography of Arthur Cayley 137 Cosets and Lagrange’s Theorem 138 Properties of Cosets 138 | Lagrange’s Theorem and Consequences 141 | An Application of Cosets to Permutation Groups 145 | The Rotation Group of a Cube and a Soccer Ball 146 Exercises 149 Computer Exercise 153 Biography of Joseph Lagrange 154 External Direct Products 155 Definition and Examples 155 | Properties of External Direct Products 156 | The Group of Units Modulo n as an External Direct Product 159 | Applications 161 Exercises 167 Computer Exercises 170 Biography of Leonard Adleman 173 Supplementary Exercises for Chapters 5–8 174 Contents Normal Subgroups and Factor Groups 178 Normal Subgroups 178 | Factor Groups 180 | Applications of Factor Groups 185 | Internal Direct Products 188 Exercises 193 Biography of Évariste Galois 199 10 Group Homomorphisms 200 Definition and Examples 200 | Properties of Homomorphisms 202 | The First Isomorphism Theorem 206 Exercises 211 Computer Exercise 216 Biography of Camille Jordan 217 11 Fundamental Theorem of Finite Abelian Groups 218 The Fundamental Theorem 218 | The Isomorphism Classes of Abelian Groups 218 | Proof of the Fundamental Theorem 223 Exercises 226 Computer Exercises 228 Supplementary Exercises for Chapters 9–11 230 PART Rings 235 12 Introduction to Rings 237 Motivation and Definition 237 | Examples of Rings 238 | Properties of Rings 239 | Subrings 240 Exercises 242 Computer Exercises 245 Biography of I N Herstein 248 13 Integral Domains 249 Definition and Examples 249 | Fields 250 | Characteristic of a Ring 252 Exercises 255 Computer Exercises 259 Biography of Nathan Jacobson 261 14 Ideals and Factor Rings 262 Ideals 262 | Factor Rings 263 | Prime Ideals and Maximal Ideals 267 Exercises 269 v vi Contents Computer Exercises 273 Biography of Richard Dedekind 274 Biography of Emmy Noether 275 Supplementary Exercises for Chapters 12–14 276 15 Ring Homomorphisms 280 Definition and Examples 280 | Properties of Ring Homomorphisms 283 | The Field of Quotients 285 Exercises 287 16 Polynomial Rings 293 Notation and Terminology 293 | The Division Algorithm and Consequences 296 Exercises 300 Biography of Saunders Mac Lane 304 17 Factorization of Polynomials 305 Reducibility Tests 305 | Irreducibility Tests 308 | Unique Factorization in Z[x] 313 | Weird Dice: An Application of Unique Factorization 314 Exercises 316 Computer Exercises 319 Biography of Serge Lang 321 18 Divisibility in Integral Domains 322 Irreducibles, Primes 322 | Historical Discussion of Fermat’s Last Theorem 325 | Unique Factorization Domains 328 | Euclidean Domains 331 Exercises 335 Computer Exercise 337 Biography of Sophie Germain 339 Biography of Andrew Wiles 340 Supplementary Exercises for Chapters 15–18 341 PART Fields 343 19 Vector Spaces 345 Definition and Examples 345 | Subspaces 346 | Linear Independence 347 Contents vii Exercises 349 Biography of Emil Artin 352 Biography of Olga Taussky-Todd 353 20 Extension Fields 354 The Fundamental Theorem of Field Theory 354 | Splitting Fields 356 | Zeros of an Irreducible Polynomial 362 Exercises 366 Biography of Leopold Kronecker 369 21 Algebraic Extensions 370 Characterization of Extensions 370 | Finite Extensions 372 | Properties of Algebraic Extensions 376 | Exercises 378 Biography of Irving Kaplansky 381 22 Finite Fields 382 Classification of Finite Fields 382 | Structure of Finite Fields 383 | Subfields of a Finite Field 387 Exercises 389 Computer Exercises 391 Biography of L E Dickson 392 23 Geometric Constructions 393 Historical Discussion of Geometric Constructions 393 | Constructible Numbers 394 | Angle-Trisectors and Circle-Squarers 396 Exercises 396 Supplementary Exercises for Chapters 19–23 399 PART Special Topics 401 24 Sylow Theorems 403 Conjugacy Classes 403 | The Class Equation 404 | The Probability That Two Elements Commute 405 | The Sylow Theorems 406 | Applications of Sylow Theorems 411 Exercises 414 Computer Exercise 418 Biography of Ludwig Sylow 419 viii Contents 25 Finite Simple Groups 420 Historical Background 420 | Nonsimplicity Tests 425 | The Simplicity of A5 429 | The Fields Medal 430 | The Cole Prize 430 | Exercises 431 Computer Exercises 432 Biography of Michael Aschbacher 434 Biography of Daniel Gorenstein 435 Biography of John Thompson 436 26 Generators and Relations 437 Motivation 437 | Definitions and Notation 438 | Free Group 439 | Generators and Relations 440 | Classification of Groups of Order Up to 15 444 | Characterization of Dihedral Groups 446 | Realizing the Dihedral Groups with Mirrors 447 Exercises 449 Biography of Marshall Hall, Jr 452 27 Symmetry Groups 453 Isometries 453 | Classification of Finite Plane Symmetry Groups 455 | Classification of Finite Groups of Rotations in R3 456 Exercises 458 28 Frieze Groups and Crystallographic Groups 461 The Frieze Groups 461 | The Crystallographic Groups 467 | Identification of Plane Periodic Patterns 473 Exercises 479 Biography of M C Escher 484 Biography of George Pólya 485 Biography of John H Conway 486 29 Symmetry and Counting 487 Motivation 487 | Burnside’s Theorem 488 | Applications 490 | Group Action 493 Exercises 494 Biography of William Burnside 497 30 Cayley Digraphs of Groups 498 Motivation 498 | The Cayley Digraph of a Group 498 | Hamiltonian Circuits and Paths 502 | Some Applications 508 Contents Exercises 511 Biography of William Rowan Hamilton 516 Biography of Paul Erdös 517 31 Introduction to Algebraic Coding Theory 518 Motivation 518 | Linear Codes 523 | Parity-Check Matrix Decoding 528 | Coset Decoding 531 | Historical Note: The Ubiquitous Reed-Solomon Codes 535 Exercises 537 Biography of Richard W Hamming 542 Biography of Jessie MacWilliams 543 Biography of Vera Pless 544 32 An Introduction to Galois Theory 545 Fundamental Theorem of Galois Theory 545 | Solvability of Polynomials by Radicals 552 | Insolvability of a Quintic 556 Exercises 557 Biography of Philip Hall 560 33 Cyclotomic Extensions 561 Motivation 561 | Cyclotomic Polynomials 562 | The Constructible Regular n-gons 566 Exercises 568 Computer Exercise 569 Biography of Carl Friedrich Gauss 570 Biography of Manjul Bhargava 571 Supplementary Exercises for Chapters 24–33 572 Selected Answers A1 Text Credits A40 Photo Credits A42 Index of Mathematicians A43 Index of Terms A45 ix Index of Mathematicians (Biographies appear on pages in boldface.) Abbati, Pietro, 141 Abel, Niels, 32, 39, 325, 419, 552 Adleman, L., 162, 164, 165, 173–174 Allenby, R., 123 Artin, Emil, 321, 352, 353 Artin, Michael, 352 Aschbacher, Michael, 424, 431, 434, 452 Bell, E.T., 383 Berlekamp, Elwyn, 536 Bhargava, Manjul, 571 Bieberbach, L., 476 Birkhoff, Garrett, 304 Boole, George, 245 Brauer, Richard, 421, 431 Burnside, William, 421, 488, 497, 560 Eisenstein, Ferdinard, 309, 570 Erdös, Paul, 517 Escher, M.C., 484, 485, 509–511 Euclid, 6, 393 Euler, Leonhard, 39, 44, 154, 325, 339, 370, 517 Feit, Walter, 421, 424, 431, 436, 497, 553 Fermat, Pierre, 143, 325, 326 Fields, John, 430 Fischer, Bernd, 424 Fourier, Joseph, 199 Fraenkel, Abraham, 237 Frobenius, Georg, 174, 408, 488 Cauchy, Augustin-Louis, 98, 104, 121, 187, 199, 266, 325, 354, 408 Cayley, Arthur, 31, 89, 90, 95, 126, 137, 382, 444, 498 Chevalley, Claude, 431 Cole, Frank, 421, 430 Conway, John H., 424, 486 Courant, Richard, 353 Crowe, Donald, 467 Galois, Evariste, 40, 123, 138, 154, 178, 199, 369, 382, 383, 420, 421, 429, 552, 553, 566, 568, 570, 571 Gauss, Carl, 39, 112, 121, 160, 274, 307, 309, 313, 325, 334, 339, 378, 383, 394, 561, 564, 567, 568, 570, 571 Germain, Sophie, 339 Gersonides, 282 Gorenstein, Daniel, 423, 435, 534 Griess, Robert, 424 Davenport, Harold, 452 da Vinci, Leonardo, 455 Dedekind, Richard, 237, 274, 383, 570 De Morgan, Augustus, 12, 90 de Seguier, J A., 62 Dickson, L.E., 392, 421, 431 Dirichlet, Peter, 325, 570 Dyck, Walther, 40, 442 Hall, Marshall, 434, 452 Hall, Philip, 452, 560 Hamilton, William Rowen, 196, 502, 516 Hamming, Richard, 520, 521, 535, 542 Hardy, G.H., 452 Hermite, Charles, 370 Herstein, I.N., 248, 435 A43 A44 INDEX OF MATHEMATICIANS Hilbert, David, 237, 275, 325, 352, 353, 475 Hölder, Otto, 131, 180, 217, 420, 430 Holst, Elling, 419 Pascal, Blaise, 12 Pless, Vera, 544 Poincaré, Henri, 369, 451 Pólya, George, 484, 485 Jacobson, Nathan, 261 Jordan, Camille, 180, 200, 207, 217, 383, 419, 420, 421, 429 Rankin, R.A., 509 Reed, Irving, 535, 536 Riemann, Bernhard, 570 Rivest, R., 162, 164, 173 Ruffini, Paolo, 102, 142 Kaplansky, Irving, 381 Klein, Felix, 275, 429, 430 Kline, Morris, 274 Knuth, Donald, 434, 452 Kronecker, Leopold, 218, 325, 354, 369 Kummer, Ernst, 325, 326, 369, 570 Lagrange, Joseph-Louis, 39, 121, 141, 154, 199, 339 Lamé, Gabriel, 325, 326 Landau, Edmund, 326 Lange, Serge, 321 Laplace, Pierre-Simon, 121, 154 Larson, Loren, 108 Legendre, Adrien-Marie, 199, 325 Lie, Sophus, 419 Lindemann, Ferdinand, 360, 369, 370, 394 Liouville, Joseph, 370 Mac Lane, Saunders, 304 MacWilliams, Jessie, 543 Mathieu, Emile, 421 Maurolycus, Francisco, 12 McElicee, Robert, 452, 535, 536 Miller, G.A., 138, 174, 421 Miyaoka, Yoichi, 326 Moore, E.H., 131, 383 Motzkin, T., 333 Netto, Eugen, 89 Newton, Isaac, 39, 154, 339 Noether, Emmy, 275, 330, 556 Schattschneider, Doris, 467 Shamir, Adi, 162, 164, 165, 173 Shannon, Claude, 535 Singer, Richard, 331 Slepian, David, 531 Sloane, Neil, 543 Smith, Stephen, 424 Solomon, Gustave, 535, 536 Steinitz, Ernst, 285, 375 Sylow, Ludwig, 91, 406, 419 Sylvester, J.J., 89–90, 137 Taylor, Richard, 326, 340 Taussky-Todd, Olga, 353 Thompson, John G., 421, 423, 424, 431, 435, 436, 452, 453, 497, 556, 560 van der Waerden, B.L., 217 Verhoeff, J., 110, 111 Wantzel, Pierre, 394 Weber, Heinrich, 40, 44, 251 Weyl, Hermann, 275, 455 Wiles, Andrew, 326, 340, 430, 571 Zariski, Oscar, 435 Zelmanov, Efim, 497 Zierler, N., 534 Zorn, Max, 248 Index of Terms Abel Prize, 39, 436 Abelian group, 32, 41 Addition modulo n, Additive group of integers modulo n, 42 Algebraic closure, 377, 378 element, 370 extension, 370 Algebraically closed field, 378 Alternating group, 106 Annihilator, 272 Arc, 498 Ascending chain condition, 329, 336 Associates, 322 Associativity, 32, 41 Automorphism(s) Frobenius, 389 group, 131, 509 group of E over F, 546 inner, 130 of a group, 130 Axioms for a group, 41 for a ring, 237 for a vector space, 345 Basis for a vector space, 347 Binary code, 523 operation, 40 strings, 161 Boolean ring, 245 Burnside’s Theorem, 489 Cancellation property for groups, 48 property for integral domains, 250 Cauchy’s Theorem, 187, 408 Cayley digraph, 498 Cayley table, 31 Cayley’s Theorem, 126 generalized, 426 Center of a group, 62 of a ring, 243 Centralizer of an element, 64 of a subgroup, 66 Characteristic of a ring, 252 Characteristic subgroup, 174 Check digit, Check-digit scheme, 110 Chinese Remainder Theorem for Rings, 341 Circle in F, 394 Class equation, 404 Closure, 31, 40 Code binary, 523 dual of, 573 Hamming, 520 (n,k) linear, 523 self-dual, 573 systematic, 526 ternary, 524 word, 520, 523 Cole Prize, 321, 352, 392, 424, 430, 434, 436, 571 Commutative diagram, 208 Commutative operation, 32 Commutator subgroup, 174 Composition of functions, 19 Composition factors, 420 Conjugacy class, 91, 403 Conjugate elements, 403 subgroups, 91, 408 Conjugation, 126 Constant polynomial, 295 Constructible number, 394 Constructible regular n-gons, 566 Content of a polynomial, 306 Coset decoding, 531 A45 A46 INDEX OF TERMS leader, 531 left, 138 representative, 138 right, 138 Crystallographic groups, 467 Crystallographic restriction, 473 Cube, rotation group of, 147 Cycle m-, 98 notation, 98 Cyclic group, 72 rotation group, 34 subgroup, 61 Cyclotomic extension, 562 polynomial, 310, 562 Decoding coset, 531 maximum-likelihood, 518 nearest neighbor, 520 parity-check matrix, 528 Degree of a over F, 372 of an extension, 372 of a polynomial, 295 rule, 301 DeMoivre’s Theorem, 13 Derivative, 362 Determinant, 43 Diagonal of G % G, 168 Digital signatures, 165 Dihedral groups, 31, 32 Dimension of a vector space, 349 Direct product of groups external, 155 internal, 188, 190 Direct sum of groups, 192 of rings, 239 Dirichlet’s Theorem, 228 Discrete frieze group, 461 Distance between vectors, 524 Divides, 238, 298 Division algorithm for F[x], 296 for Z, Divisor, Domain Euclidean, 331 integral, 249 Noetherian, 330 unique factorization, 328 Doubling the cube, 393, 395 Dual code, 573 Eisenstein’s criterion, 309 Element(s) algebraic, 370 conjugate, 403 degree of, 372 fixed by f, 489 idempotent, 255 identity, 31, 41, 238 inverse, 31, 41 nilpotent, 255 order of, 57 primitive, 376 square, 195 transcendental, 370 Embedding Theorem, 427 Empty word, 438 Equivalence class, 16 Equivalence relation, 16 Equivalent under group action, 487 Euclidean domain, 331 Euclid’s Lemma, generalization of, 23 Euler phi-function, 79 Even permutation, 105 Exponent of a group, 175 Extension algebraic, 370 cyclotomic, 562 degree, 372 field, 354 finite, 372 infinite, 372 simple, 370 transcendental, 370 External direct product, 155 Factor group, 180 of a ring element, 238 ring, 263 Factor Theorem, 298 INDEX OF TERMS Feit-Thompson Theorem, 421, 423, 436, 497, 553 Fermat prime, 568 Fermat’s Last Theorem, 325–327 Fermat’s Little Theorem, 143 Field algebraic closure of, 377, 378 algebrically closed, 378 definition of, 250 extension, 354 fixed, 546 Galois, 383 of quotients, 285 perfect, 364 splitting, 356 Fields Medal, 423, 430, 436, 497 Finite dimensional vector space, 349 Finite extension, 372 First Isomorphism Theorem for groups, 207 for rings, 283 Fixed field, 546 Free group, 439 Frieze pattern, 461 Frobenius map, 289, 389 Function composition, 19 definition of, 18 domain, 18 image under, 18 one-to-one, 19 onto, 20 range, 18 Fundamental region, 473 Fundamental Theorem of Algebra, 378 of Arithmetic, of Cyclic Groups, 77 of Field Theory, 354 of Finite Abelian Groups, 218 of Galois Theory, 550 of Group Homomorphisms, 207 of Ring Homomorphisms, 284 GAP, 109 G/Z Theorem, 186 Galois field, 383 group, 546, 558 Gaussian integers, 241, 332 A47 Gauss’s Lemma, 307 Generating region of a pattern, 473 Generator(s) of a cyclic group, 61, 72 of a group, 47 in a presentation, 441 Geometric constructions, 393 Glide-reflection, 454 nontrivial, 464 trivial, 464 Greatest common divisor, Group Abelian, 32, 41 action, 493 alternating, 106 automorphism, 131, 509 automorphism of, 130 center of, 62 color graph of a , 499 commutative, 32 composition factors, 420 crystallographic, 467 cyclic 34, 61, 72 definition, 41 dicyclic, 445, 450 dihedral, 31, 32 discrete frieze group, 461 factor, 180 finite, 57 free, 439 frieze, 461 Galois, 546, 558 general linear, 43 generator(s), 47, 61, 72, 441 Hamiltonian, 514 Heisenberg, 54 homomorphism of, 200 icosahedral, 430, 457 infinite dihedral, 446 inner automorphism, 131 integers mod n, 42 isomorphic, 123 isomorphism, 123 non-Abelian, 32, 41 octahedral, 457 order of, 57 p- 404, 417 permutation, 95 presentation, 441 quarternions, 91, 196, 442 A48 INDEX OF TERMS quotient, 180 representation, 211 simple, 420 solvable, 553 space, 475 special linear, 45 symmetric, 97 symmetry, 33, 34, 453 tetrahedral, 457 of units, 243 wallpaper, 467 Half-turn, 463 Hamiltonian circuit, 503 group, 514 path, 503 Hamming code, 520 distance, 524 weight of a code, 524 weight of a vector, 524 Homomorphism(s) Fundamental Theorem of, 207, 284 kernel of, 200 of a group, 200 natural, 210, 284 of a ring, 280 Ideal annihilator, 272 definition of, 262 finitely generated, 336 generated by, 263 maximal, 267 nil radical of, 272 prime, 267 principal, 263 product of, 270 proper, 262 sum of, 270 test, 262 trivial, 263 Idempotent, 255 Identity element, 31, 41, 238 Index of a subgroup, 142 Index Theorem, 426 Induction first principle of, 13 second principle of, 14 Inner automorphism, 130 Integral domain, 249 Internal direct product, 188, 190 International standard book number, 24 Inverse element, 31, 41 Inverse image, 204 Inversion, 135 Irreducibility tests, 306, 308 Irreducible element, 322 Irreducible polynomial, 305 ISBN, 24 Isometry, 453 Isomorphism(s) class, 218 First Theorem for groups, 207 First Theorem for rings, 283 of groups, 123 of rings, 280 Second Theorem for groups, 214 Second Theorem for rings, 341 Third Theorem for groups, 214 Third Theorem for rings, 341 Kernel of a homomorphism, 200 of a linear transformation, 351 Key, 162 Kronecker’s Theorem, 354 Lagrange’s Theorem 141 Latin square, 53 Lattice diagram, 80 of points, 473 unit, 473 Leading coefficient, 295 Least common multiple, Left regular representation, 127 Line in F, 394 Linear code, 523 combination, 347 transformation, 351 Linearly dependent vectors, 347 Linearly independent vectors, 347 Mathematical induction First Principle, 13 Second Principle, 14 Mapping, 18 INDEX OF TERMS Matrix addition, 42 determinant of, 43 multiplication, 43 standard generator, 526 Maximal, ideal, 267 subgroup, 232 Maximum-likelihood decoding, 518 Measure, 331 Minimal polynomial, 371 Mirror, 454 Mod p Irreducibility Test, 308 Modular arithmetic, Monic polynomial, 295 Monster, 424, 556 Multiple, Multiple zeros, 363 Multiplication modulo n, Multiplicity of a zero, 298 Natural homomorphism, 210, 281, 284 Natural mapping, 208 N/C Theorem, 209 Nearest-neighbor decoding, 520 Nilpotent element, 255 Nil radical, 272 Noetherian domain, 330 Norm, 323 Normal subgroup, 178 Normal Subgroup Test, 179 Normalizer, 91 Odd permutation, 105 Operation associative, 41 binary, 40 commutative, 32 preserving mapping, 123 table, 31 Opposite isometry, 454 Orbit of a point, 145 Orbit-Stabilizer Theorem, 146 Order or a group, 57 of an element, 57 Orthogonality Relation, 530 PID, 299 Parity-check matrix, 528 Partition of a set, 17 of an integer, 219 Perfect field, 364 Permutation definition of, 95 even, 105 group, 95 odd, 105 p-group 404, 417 Phi-function, Euler, 79 Plane of F, 394 Plane symmetry, 33 Polynomial(s) alternating, 106 constant, 295 content of, 306 cyclotomic, 310, 562 degree of, 295 derivative of, 362 Galois group of, 558 irreducible, 305 leading coefficient of, 295 minimal, 371 monic, 295 primitive, 306 reducible, 305 relatively prime, 303 ring of, 293 splits, 356 symmetric, 106 zero of, 298 Prime element of a domain, 322 ideal, 267 integer, relatively, 5, 303 subfield, 285 Primitive element, 376 Element Theorem, 375 nth root of unity 299, 562 polynomial, 306 Principal ideal domain 271, 299 Principal ideal ring, 290 Projection, 212 Proper ideal, 262 Proper subgroup, 58 Pullback, 204 A49 A50 INDEX OF TERMS Quaternions, 91, 196, 442 Quotient, 4, 297 Quotient group, 180 Quotients, field of, 285 Range, 18 Rational Root Theorem, 318 Reducible polynomial, 305 Reflection, 34, 454 Relation equivalence, 16 in a presentation, 441 Relatively prime, 5, 303 Remainder, 4, 297 Remainder Theorem, 298 Ring(s) Boolean, 245 center of, 243 characteristic of, 252 commutative, 238 definition of, 237 direct sum of, 239 factor, 263 homomorphism of, 280 isomorphism of, 280 of polynomials, 293 with unity, 238 RSA public encryption, 164 Rubik’s cube, 110 Scalar, 345 Scalar multiplication, 345 Self dual code, 573 Sicherman dice, 315 Simple extension, 370 Simple group, 420 Socks-Shoes Property, 50 Solvable by radicals, 552 Solvable group, 553 Spanning set, 347 Splitting field, 356 Squaring the circle, 393, 396 Stabilizer of a point, 115, 145 Standard array, 531 Standard generator matrix, 526 Subcode, 537 Subfield Test, 256 Subgroup(s) centralizer, 66 characteristic, 174 commutator, 174 conjugate, 91, 408 cyclic, 61 definition of, 58 diagonal, 168 Finite Test, 61 generated by a, 61 index of, 142 lattice, 80 maximal, 232 nontrivial, 58 normal, 178 One-Step Test, 59 proper, 58 Sylow p- , 407 torsion, 92 trivial, 58 Two-Step Test, 60 Subring definition of, 240 Test, 240 Trivial, 241 Subspace, 346 Subspace spanned by vectors, 347 Subspace Test, 349 Sylow p-subgroup, 407 Sylow test for nonsimplicity, 425 Sylow Theorems, 406, 408, 409 Symmetric group, 97 Symmetries of a square, 29 Symmetry group, 33, 34, 453 Syndrome of a vector, 533 Systematic code, 526 Torsion subgroup, 92 Transcendental element, 370 Transcendental extension, 370 Translation, 45, 454 Transposition, 103 Trisecting an angle, 393, 396 UFD, 328 Unique factorization domain, 328 Unique factorization theorem for a PID, 329 for D[x], 334 for F[x], 331 for Z, for Z[x], 313 in a Euclidean domain, 333 INDEX OF TERMS Unity, 238 Universal Factor Group Property, 440 Universal Mapping Property, 439 Universal Product Code, Vector, 345 Vector space basis of, 347 definition of, 345 dimension of, 349 finite dimensional, 349 infinite dimensional, 349 spanned by a set, 347 trivial, 349 Vertex of a graph, 498 Wallpaper groups, 467 Weight of a vector, 524 Weighting vector, 10 Weird dice, 315 Well-defined function, 201 Well-ordering principle, Word code, 520, 523 empty, 438 in a group, 438 Zero multiple, 363 multiplicity of, 298 of a polynomial, 298 Zero-divisor, 249 A51 This page intentionally left blank Essential Theorems in Abstract Algebra Theorem 3.1 One-Step Subgroup Test A nonempty subset H of a group G is a subgroup of G if abϪ1 is in H whenever a and b are in H Theorem 4.3 Fundamental Theorem of Cyclic Groups Every subgroup of a cyclic group is cyclic If |͗a͘| = n, then for each positive divisor k of n, ͗a͘ has exactly one subgroup of order k and no others Theorem 7.1 Lagrange’s Theorem In a finite group the order of a subgroup divides the order of the group Theorem 9.1 Normal Subgroup Test A subgroup H of G is normal in G if and only if xHxϪ1 # H for all x in G Theorem 10.3 First Isomorphism Theorem If f is a group homomorphism from G to a group, then G/Ker f Ϸ f(G) Theorem 11.1 Fundamental Theorem of Finite Abelian Groups Every finite Abelian group is a direct product of cyclic groups of prime-power order Theorem 12.3 Subring Test A nonempty subset S of a ring R is a subring if a Ϫ b and ab are in S whenever a and b are in S Theorem 13.4 Characteristic of an Integral Domain The characteristic of an integral domain is or prime Theorem 14.1 Ideal Test A nonempty subset A of a ring R is an ideal if a Ϫ b [ A whenever a and b are in A; and and ar are in A whenever a [ A and r [ R Theorem 14.4 R/A is a Field if and only if A is Maximal Let R be a communitive ring with unity and let A be an ideal of R Then R/A is a field if and only if A is maximal Theorem 15.3 First Isomorphism Theorem for Rings If f is a ring homomorphism from R to a ring, then R/Ker f Ϸ f(R) Corollary of Theorem 17.5 F[x]ր͗p(x)͘ Is a Field Let F be a field and p(x) an irreducible polynomial over F Then F[x]ր͗p(x)͘ is a field Theorem 21.5 [K : F] ϭ [K : E][E : F] If K is a finite extension field of the field E and E is a finite extension field of the field F, then [K : F] ϭ [K : E][E : F] Theorem 22.2 Structure of Finite Fields The set of nonzero elements of a finite field is a cyclic group under multiplication Theorem 24.3 Sylow’s First Theorem Let G be a finite group and p a prime If pk divides |G|, then G has a subgroup of order pk Theorem 24.5 Sylow’s Third Theorem The number of Sylow p-subgroups of G is equal to modulo p and divides |G| Furthermore, any two Sylow p-subgroups of G are conjugate Cayley Tables Cayley Table for the Dihedral Group of Order R0 R120 R240 F FЈ FЉ R0 R120 R240 F FЈ FЉ R0 R120 R240 F FЈ FЉ R120 R240 R0 FЉ F FЈ R240 R0 R120 FЈ FЉ F F FЈ FЉ R0 R120 R240 FЈ FЉ F R240 R0 R120 FЉ F FЈ R120 R240 R0 F' F" F Cayley Table for the Dihedral Group of Order R0 R90 R180 R270 H V D DЈ R0 R90 R180 R270 H V D DЈ R0 R90 R180 R270 H V D DЈ R90 R180 R270 R0 D DЈ V H R180 R270 R0 R90 V H DЈ D R270 R0 R90 R180 DЈ D H V H DЈ V D R0 R180 R270 R90 V D H DЈ R180 R0 R90 R270 D H DЈ V R90 R270 R0 R180 DЈ V D H R270 R90 R180 R0 D V D' H Notations (The number after the item indicates the page where the notation is defined.) SET THEORY SPECIAL SETS >i[ISi