**Numbers, Polynomials, and Factoring **

The Natural Numbers

The Integers

Modular Arithmetic

Polynomials with Rational Coefficients

Factorization of Polynomials

Section I in a Nutshell

**Rings, Domains, and Fields **

Rings

Subrings and Unity

Integral Domains and Fields

Ideals

Polynomials over a Field

Section II in a Nutshell

**Ring Homomorphisms and Ideals **

Ring Homomorphisms

The Kernel

Rings of Cosets

The Isomorphism Theorem for Rings

Maximal and Prime Ideals

The Chinese Remainder Theorem

Section III in a Nutshell

**Groups **

Symmetries of Geometric Figures

Permutations

Abstract Groups

Subgroups

Cyclic Groups

Section IV in a Nutshell

**Group Homomorphisms **

Group Homomorphisms

Structure and Representation

Cosets and Lagrange's Theorem

Groups of Cosets

The Isomorphism Theorem for Groups

Section V in a Nutshell

**Topics from Group Theory**

The Alternating Groups

Sylow Theory: The Preliminaries

Sylow Theory: The Theorems

Solvable Groups

Section VI in a Nutshell

**Unique Factorization **

Quadratic Extensions of the Integers

Factorization

Unique Factorization

Polynomials with Integer Coefficients

Euclidean Domains

Section VII in a Nutshell

**Constructibility Problems **

Constructions with Compass and Straightedge

Constructibility and Quadratic Field Extensions

The Impossibility of Certain Constructions

Section VIII in a Nutshell

**Vector Spaces and Field Extensions **

Vector Spaces I

Vector Spaces II

Field Extensions and Kronecker's Theorem

Algebraic Field Extensions

Finite Extensions and Constructibility Revisited

Section IX in a Nutshell

**Galois Theory **

The Splitting Field

Finite Fields

Galois Groups

The Fundamental Theorem of Galois Theory

Solving Polynomials by Radicals

Section X in a Nutshell

**Hints and Solutions **

**Guide to Notation **

**Index**