Highly regarded by instructors in past editions for its sequencing of topics and extensive set of exercises, the latest edition of Abstract Algebra retains its concrete approach with its gentle introduction to basic background material and its gradual increase in the level of sophistication as the student progresses through the book. Abstract concepts are introduced only after a careful study of important examples. Beachy and Blair’s clear narrative presentation responds to the needs of inexperienced students who stumble over proof writing, who understand definitions and theorems but cannot do the problems, and who want more examples that tie into their previous experience. The authors introduce chapters by indicating why the material is important and, at the same time, relating the new material to things from the student’s background and linking the subject matter of the chapter to the broader picture.
The fourth edition includes a new chapter of selected topics in group theory: nilpotent groups, semidirect products, the classification of groups of small order, and an application of groups to the geometry of the plane.
Students can download solutions to selected problems here.
Author(s): John A. Beachy; William D. Blair
Edition: 4
Publisher: Waveland Press Inc
Year: 2019
Language: English
Pages: 541
City: Long Grove, IL
Tags: Abstract, Algebra, John A. Beachy, William D. Blair, Math, Abstract
1 INTEGERS 1
1.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4 Integers Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2 FUNCTIONS 51
2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3 GROUPS 91
3.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3 Constructing Examples . . . . . . . . . . . . . . . . . . . . . . . 122
3.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.6 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 161
iii
iv CONTENTS
3.8 Cosets, Normal Subgroups, and Factor Groups . . . . . . . . . . . 173
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4 POLYNOMIALS 189
4.1 Fields; Roots of Polynomials . . . . . . . . . . . . . . . . . . . . 189
4.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.3 Existence of Roots . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.4 Polynomials over Z, Q, R, and C . . . . . . . . . . . . . . . . . . 221
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5 COMMUTATIVE RINGS 235
5.1 Commutative Rings; Integral Domains . . . . . . . . . . . . . . . 236
5.2 Ring Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 248
5.3 Ideals and Factor Rings . . . . . . . . . . . . . . . . . . . . . . . 262
5.4 Quotient Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
6 FIELDS 281
6.1 Algebraic Elements . . . . . . . . . . . . . . . . . . . . . . . . . 282
6.2 Finite and Algebraic Extensions . . . . . . . . . . . . . . . . . . 289
6.3 Geometric Constructions . . . . . . . . . . . . . . . . . . . . . . 295
6.4 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.5 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
6.6 Irreducible Polynomials over Finite Fields . . . . . . . . . . . . . 313
6.7 Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . 320
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
7 STRUCTURE OF GROUPS 329
7.1 Isomorphism Theorems; Automorphisms . . . . . . . . . . . . . 330
7.2 Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
7.3 Groups Acting on Sets . . . . . . . . . . . . . . . . . . . . . . . 345
7.4 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . 353
7.5 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . 358
7.6 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
7.7 Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
8 GALOIS THEORY 383
8.1 The Galois Group of a Polynomial . . . . . . . . . . . . . . . . . 384
8.2 Multiplicity of Roots . . . . . . . . . . . . . . . . . . . . . . . . 390
8.3 The Fundamental Theorem of Galois Theory . . . . . . . . . . . 394
8.4 Solvability by Radicals . . . . . . . . . . . . . . . . . . . . . . . 405
8.5 Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . . . . . 411
8.6 Computing Galois Groups . . . . . . . . . . . . . . . . . . . . . 417
CONTENTS v
9 UNIQUE FACTORIZATION 427
9.1 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . . 428
9.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . . . 435
9.3 Some Diophantine Equations . . . . . . . . . . . . . . . . . . . . 441
10 GROUPS: SELECTED TOPICS 453
10.1 Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 453
10.2 Internal Semidirect Products of Groups . . . . . . . . . . . . . . 457
10.3 External Semidirect Products of Groups . . . . . . . . . . . . . . 463
10.4 Classification of Groups of Small Order . . . . . . . . . . . . . . 471
10.5 The Orthogonal Group O2.R/ . . . . . . . . . . . . . . . . . . . 477
10.6 Isometries of R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
APPENDIX 487
A.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
A.2 Construction of the Number Systems . . . . . . . . . . . . . . . . 490
A.3 Basic Properties of the Integers . . . . . . . . . . . . . . . . . . . 493
A.4 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
A.5 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 498
A.6 Solution of Cubic and Quartic Equations . . . . . . . . . . . . . . 504
A.7 Dimension of a Vector Space . . . . . . . . . . . . . . . . . . . . 512
BIBLIOGRAPHY 516
SELECTED ANSWERS 518
INDEX OF SYMBOLS 527
INDEX 531
