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.
Table of Contents
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
". . .these two volumes constitute an outstanding and unique text on algebra and its various applications, which is matchlessly rich in content, comprehensive and well-arranged. . .[a] masterpiece. . . .the mathematical community should have good reason to be grateful to the author for having added a new standard text to the existing literature. . .with many distinctive and unique features, and. . .comparable to the celebrated standard texts. . ..Due to its peerless width of the material. . .this textbook. . .is certainly predestined to become an indispensable tool kit for everyone needing and using algebraic methods. "
---Zentralblatt fur Mathematik