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.

chapter 1|9 pages

Introduction: The Art of Doing Arithmetic

chapter 2|19 pages

Rings and ring homomorphisms

chapter 3|25 pages

Integral domains and fields

chapter 4|22 pages

Polynomial and power series rings

chapter 5|18 pages

Ideals and quotient rings

chapter 6|18 pages

Ideals in commutative rings

chapter 7|23 pages

Factorization in integral domains

chapter 10|18 pages

Modules and integral ring extensions

chapter 11|14 pages

Noetherian rings

chapter 12|31 pages

Field extensions

chapter 13|20 pages

Splitting fields and normal extensions

chapter 14|23 pages

Separability of field extensions

chapter 15|10 pages

Field theory and integral ring extensions

chapter 16|17 pages

Affine algebras

chapter 17|29 pages

Ring theory and algebraic geometry

chapter 18|18 pages


chapter 19|14 pages

Factorization of ideals

chapter 21|16 pages

The Galois group of a field extension

chapter 22|31 pages

Algebraic Galois extensions

chapter 23|20 pages

The Galois group of a polynomial

chapter 24|21 pages

Roots of unity and cyclotomic polynomials

chapter 25|14 pages

Pure equations and cyclic extensions

chapter 26|25 pages

Solvable equations and radical extensions