A First Course in Abstract Algebra

by Fraleigh, John B.

• ISBN13: 9780201534672
• ISBN10: 0201534673
• Edition: 5th
• Format: Hardcover
• Copyright: 1994-01-01
• Publisher: ADDISON WESLEY LONGMAN INC

Summary

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

Table of Contents

Sets and Relations

Groups and Subgroups

Introduction and Examples

Binary Operations

Isomorphic Binary Structures

Groups

Subgroups

Cyclic Groups

Generators and Cayley Digraphs

Permutations, Cosets, and Direct Products

Groups of Permutations

Orbits, Cycles, and the Alternating Groups

Cosets and the Theorem of Lagrange

Direct Products and Finitely Generated Abelian Groups

Plane Isometries

Homomorphisms and Factor Groups

Homomorphisms

Factor Groups

Factor-Group Computations and Simple Groups

Group Action on a Set

Applications of G-Sets to Counting

Rings and Fields

Rings and Fields

Integral Domains

Fermat's and Euler's Theorems

The Field of Quotients of an Integral Domain

Rings of Polynomials

Factorization of Polynomials over a Field

Noncommutative Examples

Ordered Rings and Fields

Ideals and Factor Rings

Homomorphisms and Factor Rings

Prime and Maximal Ideas

Grouml;bner Bases for Ideals

Extension Fields

Introduction to Extension Fields

Vector Spaces

Algebraic Extensions

Geometric Constructions

Finite Fields

Advanced Group Theory

Isomorphism Theorems

Series of Groups

Sylow Theorems

Applications of the Sylow Theory

Free Abelian Groups

Free Groups

Group Presentations

Groups in Topology

Simplicial Complexes and Homology Groups

Computations of Homology Groups

More Homology Computations and Applications

Homological Algebra

Factorization

Unique Factorization Domains

Euclidean Domains

Gaussian Integers and Multiplicative Norms

Automorphisms and Galois Theory

Automorphisms of Fields

The Isomorphism Extension Theorem

Splitting Fields

Separable Extensions

Totally Inseparable Extensions

Galois Theory

Illustrations of Galois Theory

Cyclotomic Extensions

Insolvability of the Quintic

Appendix: Matrix Algebra

Notations

Answers to odd-numbered exercises not asking for definitions or proofs

Index

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