Fundamentals of Abstract Algebra 0070400350, 9780070400351

Table of contents :
Cover page
Title page
Contents
1. Sets, Relations, and Integers
1.1 Sets
1.2 Integers
1.3 Relations
1.4 Partially Ordered Sets
1.5 Functions
1.6 Binary Operations
2. Introduction to Groups
2.1 Elementary Properties of Groups
3. Permutation Groups
3.1 Permutation Groups
4. Subgroups and Normal Subgroups
4.1 Subgroups
4.2 Cyclic Groups
4.3 Lagrange's Theorem
4.4 Normal Subgroups and Quotient Groups
5. Homomorphism and Isomorphisms of Groups
5.1 Homomorphisms of Groups
5.2 Isomorphism and Correspodece Theorems
5.3 The Groups D4 and Q8
5.4 Group Actions
6. Direct Product of Groups
6.1 External and Internal Direct Product
7. Sylow Theorem
7.1 Conjugacy Classes
7.2 Cauchy's Theorem and p-groups
7.3 Sylow Theorems
7.4 Some Applications of Sylow Theorems
8. Solvable and Nilpotent Groups
8.1 Solvable Groups
8.2 Nilpotent Groups
9. Finitely Generated Abelian Groups
9.1 Finite Abelian Groups
9.2 Finitely Generated Abelian Groups
10. Introduction to Rings
10.1 Elementary Properties
10.2 Some Important Rings
11. Subrings, Ideals, and Homomorphisms
11.1 Subrings and Subfields
11.2 Ideals and Quotient Rings
11.3 Homomorphisms and Isomorphisms
12. Ring Embeddings
12.1 Embedding of Rings
13. Direct Sum of Rings
13.1 Complete Direct Sum and Direct Sum
14. Polynomial Rings
14.1 Polynomial Rings
15. Euclidean Domain
15.1 Euclidean Domains
15.2 Greatest Common Divisors
15.3 Prime and Irreducible Elements
16. Unique Factorization Domains
16.1 Unique Factorization Domains
16.2 Factorization of Polynomials over a UFD
16.3 Irreducibility of Polynomials
17. Maximal, Prime, and Primary Ideals
17.1 Maximal, Prime, and Primary Ideals
17.2 Jacobson Semisimple Ring
18. Noetherian and Artinian Rings
18.1 Noetherian and Artinian Rings
19. Modules and Vector Spaces
19.1 Modules and Vector Spaces
20. Rings of Matrices
20.1 Full Matrix Rings
20.2 Rings of Triangular Matrices
21. Field Extensions
21.1 Algebraic Extensions
21.2 Splitting Fields
21.3 Algebraically Closed Fields
22. Multiplicity of Roots
22.1 Multiplicity of Roots
23. Finite Fields
23.1 Finite Fields
24. Galois Theory and Applications
24.1 Normal Extensions
24.2 Galois Theory
24.3 Roots of Unity and Cyclotomic Polynomials
24.4 Solvability of Polynomials by Radicals
25. Geometric Constructions
25.1 Geometric Constructions
26. Coding Theory
26.1 Binary Codes
26.2 Polynomial and Cyclic Codes
26.3 Bose-Chauduri-Hocquenghem Codes
27. Grobner Bases
27.1 Affine Varieties
27.2 Grobner Bases
Selected Bibliography
Answers and Hints to Selected Exercises
Index