Includes bibliographical references and indexes.

Rings And Fields Introduction: The Art of Doing Arithmetic Rings and Ring Homomorphisms Integral Domains and Fields Polynomial and Power Series Rings Ideals and Quotient Rings Ideals in Commutative Rings Factorization in Integral Domains Factorization in Polynomial and Power Series Rings Number-Theoretical Applications of Unique Factorization Modules Noetherian Rings Field Extensions Splitting Fields and Normal Extensions Separability of Field Extensions Field Theory and Integral Ring Extensions Affine Algebras Ring Theory and Algebraic Geometry Localization Factorization of Ideals Introduction to Galois Theory: Solving Polynomial Equations The Galois Group of a Field Extension Algebraic Galois Extensions The Galois Group of a Polynomial Roots of Unity and Cyclotomic Polynomials Pure Equations and Cyclic Extensions Solvable Equations and Radical Extensions Epilogue: The Idea of Lie Theory as a Galois Theory for Differential Equations Bibliography

This is a comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.