An Introduction to Abstract Algebra

Description

Abstract algebra is a cornerstone to modern mathematics. Other areas of mathematics heavily depend upon abstract algebra, and abstract algebra is found in a multitude of disciplines. The goal of this textbook is to be a source for a first undergraduate course in abstract algebra. Topics progress from the structure of mathematical proof, to groups, fields, and then rings. The universal nature of abstract algebra is illustrated in the textbook by the demonstration that certain geometric constructions are impossible.

Content

1. Preliminaries
   1. Introduction to Abstract Algebra
   2. Logic and Proof
   3. Set Theory
   4. Mappings and Equivalence Relations
2. Group Theory
   1. Binary Operations
   2. Introduction to Groups
   3. Cyclic Groups
   4. Dihedral Groups
   5. Groups of Permutations
   6. Alternating Groups
   7. Subgroups
   8. Homomorphisms and Isomorphisms
   9. Cosets and Normal Subgroups
   10. Quotient Groups
   11. Direct Products
   12. Catalog of Finite Groups
3. Field Theory
   1. Introduction to Fields
   2. Polynomials
   3. Irreduciblity
   4. Vector Spaces
   5. Extension Fields
   6. Algebraic Extensions
   7. Geometric Constructions
4. Ring Theory
   1. Introduction to Rings
   2. Integral Domains
   3. Ideals
5. Bibliography