An introduction to abstract algebra. Provides valuable experience for any further axiomatic study of mathematics. DLC: Algebra, Abstract.

Table of Contents

0 Sets and Relations

I Groups and Subgroups

1 Binary Operations

2 Groups

3 Abelian Examples

4 Nonabelian Examples

5 Subgroups

6 Cyclic Groups

7 Generating Sets and Cayley Digraphs

II Structure of Groups

8 Groups of Permutations

9 Finitely Generated Abelian Groups

10 Cosets and the Theorem of Lagrange

11 Plane lsometries

Ill Homomorphisms and Factor Groups

12 Factor Groups

13 Factor-Group Computations and Simple Groups

14 Group Action on a Set

15 Applications of G-Sets to Counting

IV Advanced Group Theory

16 Isomorphism Theorems

17 Sylow Theorems

18 Series of Groups

19 Free Abelian Groups

20 Free Groups

21 Group Presentations

V Rings and Fields

22 Rings and Fields

23 Integral Domains

24 Fermat's and Euler's Theorems

25 Encryption

VI Constructing Rings and Fields

26 the Field of Quotients of an Integral Domain

27 Rings of Polynomials

28 Factorization of Polynomials over a Field

29 Algebraic Coding Theory

30 Homomorphisms and Factor Rings

31 Prime and Maximal Ideals

32 Noncommutative Examples

VII Commutative Algebra

33 Vector Spaces

34 Unique Factorization Domains

35 Euclidean Domains

36 Number Theory

37 Algebraic Geometry

38 Grabner Bases for Ideals

VI 11 Extension Fields

39 Introduction to Extension Fields

40 Algebraic Extensions

41 Geometric Constructions

42 Finite Fields

IX Galois Theory

43 Introduction to Galois Theory

44 Splitting Fields

45 Separable Extensions

46 Galois Theory

47 Illustrations of Galois Theory

48 Cyclotomic Extensions

49 lnsolvability of the Quintic

Appendix: Matrix Algebra