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Contemporary Abstract Algebra by Gallian
Contemporary Abstract Algebra by Joseph A. Gallian
Used the book (twice) -- for Modern Algebra I -- Fall 2011, see also Group theory: course.
A lot of content, examples, and applications.
Contents
1 integers and equivalence relations
0 preliminaries
- Properties of integers 3
- Modular arithmetic 7
- Mathematical induction 12
- Equivalence relations 15
- Functions (mappings) 18
Part 2 groups
1 introduction to groups 29
- Symmetries of a square 29
- The dihedral groups 32
2 groups 40
- Definition and examples of groups
- Elementary properties of groups 48
3 finite groups; subgroups
- Terminology and notation 57
- Subgroup tests 58
- Examples of subgroups 61
4 cyclic groups
- Properties of cyclic groups 72
- Classification of subgroups of cyclic groups 77
- Definition and notation 95
- Cycle nation 98
- Properties of permutations 100
- A check digit scheme based on $D_5$ 110
- Motivation 122
- Definition and examples 122
- Cayley's theorem 126
- Properties of isomorphisms 128
- Automorphisms 129
7 cosets and Lagrange's theorem
- Properties of cosets 138
- Lagrange's theorem and consequences 141
- An application of cosets of permutation groups 145
- The rotation group of a cube and a soccer ball 146
8 external direct products
- Definition and examples 155
- Properties of external direct products 156
- The group of units modulo $n$ as an external direct products 159
- Applications 161
9 normal subgroups and factor groups
- Normal subgroups 178
- Factor groups 180
- Applications of factor groups 185
- Internal direct products 188
10 group homomorphisms
- Definition and examples 200
- Properties of homomorphisms 202
- The first isomorphism theorem 206
11 fundamental theorem of finite abelian groups
- The fundamental theorem 218
- The isomorphism classes of abelian groups 218
- Proof of the fundamental theorem 223
Part 3 rings
12 introduction to rings
Motivation and definition 237 Examples of rings 238 Properties of rings 239 Subrings 240
13 integral domains
Definition and examples 249 Fields 250 Characteristic of a ring 225
14 ideals and factor rings
Ideals 262 Factor rings 263 Prime ideals and maximal ideals 267
15 ring homomorphisms
Definition and example 280 Properties of ring homomorphisms 283 The field of quotients 285
16 polynomial rings
Notation and terminology 293 The division algorithm and consequences 296
17 factorization of polynomials
Reducibility tests 305 Irreducibility tests 308 Unique factorization in $z[x]$ 313 Weird dice: an application of unique factorization 314
18 divisibility in integral domains
Irreducibles, primes 322 Historical discussion of Fermat's last theorem 325 Unique factorization domains 328 Euclidean domains 331
Part 4 fields
Definition and examples 345 Subspaces 346 Linear independence 347
20 extension fields
The fundamental theorem of field theory 354 Splitting fields 356 Zeros of an irreducible polynomial 362
21 algebraic extensions
Characterization of extensions 370 Finite extensions 372 Properties of algebraic extensions 376
22 finite fields
Classification of finite fields 382 Construction of finite fields 383 Subfields of a finite field 387
23 geometric constructions
Historical discussion of geometric constructions 393 Constructible numbers 394 Angle-trisectors and circle-squarers 396
Part 5 special topics
24 Sylow theorems
Conjugacy classes 403 The class equation 404 The probability that two elements commute 405 The Sylow theorems 406 Applications of Sylow theorems 411
25 finite simple groups
Historical background 420 Nonsimplicity tests 245 The simplicity of $A_5$ 429 The fields medal 430 The Cole prize 430
26 generators and relations
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 group 446 Realizing the dihedral groups with mirrors 447
27 symmetry groups
Isometries 453 Classification of finite plane symmetry group 455 Classification of finite groups of rotations in $R^3$ 456
28 frieze groups and crystallographic groups
The frieze groups 461 The crystallographic groups 467 Identification of plane periodic patterns 473
29 symmetry and counting
Motivation 487 Burnside's theorem 488 Applications 490 Group action 493
30 Cayley digraphs of groups
Motivation 498 The Cayley digraph of a group 498 Hamiltonian circuits and paths 502 Some applications 508
31 introduction to algebraic coding theory
Motivation 518 Liner codes 523 Parity-check matrix decoding 528 Coset decoding 531 Historical note: the ubiquitous Reed-Solomon codes 535
32 an introduction to Galois theory
Fundamental theorem of Galois theory 545 Solvability of polynomials by radicals 552 Insolvability of a quintic 556 Cyclotomic extensions 561 Motivation 561 Cyclotomic polynomials 562 The constructible regular $n$-gons