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Contemporary Abstract Algebra by Gallian

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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

Part 2 groups

1 introduction to groups 29

2 groups 40

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

5 permutation groups

  • Definition and notation 95
  • Cycle nation 98
  • Properties of permutations 100
  • A check digit scheme based on $D_5$ 110

6 isomorphisms

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

10 group homomorphisms

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

19 vector spaces

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