Contemporary Abstract Algebra 9th Edition by Joseph Gallian ISBN 1305657969 9781305657960

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ISBN 10: 1305657969
ISBN 13: 9781305657960
Author: Joseph Gallian

CONTEMPORARY ABSTRACT ALGEBRA, NINTH EDITION is primarily intended for an abstract algebra course whose main purpose is to enable students to do computations and write proofs. Gallian�s text stresses the importance of obtaining a solid introduction to the traditional topics of abstract algebra, while at the same time presenting it as a contemporary and very much an active subject which is currently being used by working physicists, chemists, and computer scientists.

Contemporary Abstract Algebra 9th Edition Table of contents:

Chapter 1: Groups

• 1.1 Introduction to Groups
• 1.2 Examples of Groups
• 1.3 Properties of Groups
• 1.4 Subgroups
• 1.5 Cyclic Groups
• 1.6 Permutation Groups
• 1.7 Cosets and Lagrange’s Theorem
• 1.8 Group Homomorphisms
• 1.9 Isomorphisms
• 1.10 The Fundamental Theorem of Finite Abelian Groups
• 1.11 Group Actions
• 1.12 Sylow’s Theorems

Chapter 2: Rings

• 2.1 Introduction to Rings
• 2.2 Examples of Rings
• 2.3 Subrings and Ideals
• 2.4 Ring Homomorphisms
• 2.5 Quotient Rings
• 2.6 Division Rings and Fields
• 2.7 The Ring of Polynomials
• 2.8 The Structure of Fields
• 2.9 Field Extensions

Chapter 3: Polynomials

• 3.1 Polynomial Rings
• 3.2 The Division Algorithm
• 3.3 Roots of Polynomials
• 3.4 Factorization of Polynomials
• 3.5 The Field of Complex Numbers
• 3.6 The Fundamental Theorem of Algebra
• 3.7 Eisenstein’s Criterion

Chapter 4: Vector Spaces

• 4.1 Introduction to Vector Spaces
• 4.2 Subspaces
• 4.3 Linear Independence and Basis
• 4.4 Dimension and Rank
• 4.5 Linear Transformations
• 4.6 Eigenvalues and Eigenvectors
• 4.7 The Characteristic Polynomial

Chapter 5: Linear Transformations and Matrices

• 5.1 Introduction to Matrices
• 5.2 Matrix Operations
• 5.3 Matrix Rank
• 5.4 The Inverse of a Matrix
• 5.5 Eigenvalues and Eigenvectors
• 5.6 Diagonalization
• 5.7 Jordan Canonical Form

Chapter 6: Groups and Rings in Action

• 6.1 Applications to Geometry
• 6.2 Symmetry Groups in Geometry
• 6.3 Applications to Coding Theory
• 6.4 The Chinese Remainder Theorem
• 6.5 Applications to Cryptography

Chapter 7: Number Theory

• 7.1 Divisibility and Primes
• 7.2 Euclidean Algorithm
• 7.3 Modular Arithmetic
• 7.4 The Chinese Remainder Theorem
• 7.5 Fermat’s Little Theorem
• 7.6 The Fundamental Theorem of Arithmetic
• 7.7 Diophantine Equations

Chapter 8: Galois Theory

• 8.1 Introduction to Galois Theory
• 8.2 Field Extensions
• 8.3 The Galois Group of a Polynomial
• 8.4 Solvability by Radicals
• 8.5 The Fundamental Theorem of Galois Theory
• 8.6 Applications of Galois Theory

Chapter 9: Advanced Topics in Group Theory

• 9.1 Simple Groups
• 9.2 Solvable Groups
• 9.3 Nilpotent Groups
• 9.4 The Sylow Theorems Revisited
• 9.5 The Classification of Finite Simple Groups

Chapter 10: Introduction to Homological Algebra

• 10.1 Chain Complexes
• 10.2 Exact Sequences
• 10.3 Homology and Cohomology Groups
• 10.4 Applications to Group and Ring Theory

Appendices

• A. Review of Set Theory
• B. Review of Number Systems
• C. Solutions to Selected Exercises
• D. Index

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