ABSTRACT

Like its popular predecessors, this text develops ring theory first by drawing on students' familiarity with integers and polynomials. This unique approach motivates students in studying abstract algebra and helps them understand the power of abstraction. This edition makes it easier to teach unique factorization as an optional topic and reorganizes the core material on rings, integral domains, and fields. Along with new exercises on Galois theory, it also includes a more detailed treatment of permutations as well as new chapters on Sylow theorems.

part 1|2 pages

Part I: Numbers, Polynomials, and Factoring

chapter 1|10 pages

The Natural Numbers

chapter 2|12 pages

The Integers

chapter 3|8 pages

Modular Arithmetic

chapter 4|10 pages

Polynomials with Rational Coefficients

chapter 5|9 pages

Factorization of Polynomials

chapter |3 pages

Section I in a Nutshell

part |2 pages

Part II: Rings, Domains, and Fields

chapter 6|12 pages

Rings

chapter 7|12 pages

Subrings and Unity

chapter 8|16 pages

Integral Domains and Fields

chapter 9|12 pages

Ideals

chapter 10|11 pages

Polynomials over a Field

chapter |3 pages

Section II in a Nutshell

part 3|2 pages

Part III: Ring Homomorphisms and Ideals

chapter 11|10 pages

Ring Homomorphisms

chapter 12|8 pages

The Kernel

chapter 13|6 pages

Rings of Cosets

chapter 14|8 pages

The Isomorphism Theorem for Rings

chapter 15|12 pages

Maximal and Prime Ideals

chapter 16|9 pages

The Chinese Remainder Theorem

chapter |1 pages

Section III in a Nutshell

part 4|2 pages

Part IV: Groups

chapter 17|16 pages

Symmetries of Geometric Figures

chapter 18|10 pages

Permutations

chapter 19|12 pages

Abstract Groups

chapter 20|10 pages

Subgroups

chapter 21|12 pages

Cyclic Groups

chapter |2 pages

Section IV in a Nutshell

part 5|2 pages

Part V: Group Homomorphisms

chapter 22|10 pages

Group Homomorphisms

chapter 23|10 pages

Structure and Representation

chapter 24|10 pages

Cosets and Lagrange’s Theorem

chapter 25|8 pages

Groups of Cosets

chapter 26|8 pages

The Isomorphism Theorem for Groups

chapter |2 pages

Section V in a Nutshell

part 6|2 pages

Part VI: Topics from Group Theory

chapter 27|10 pages

The Alternating Groups

chapter 28|10 pages

Sylow Theory: The Preliminaries

chapter 29|10 pages

Sylow Theory: The Theorems

chapter 30|5 pages

Solvable Groups

chapter |3 pages

Section VI in a Nutshell

part 7|2 pages

Part VII: Unique Factorization

chapter 31|10 pages

Quadratic Extensions of the Integers

chapter 32|6 pages

Factorization

chapter 33|4 pages

Unique Factorization

chapter 34|6 pages

Polynomials with Integer Coefficients

chapter 35|7 pages

Euclidean Domains

chapter |1 pages

Section VII in a Nutshell

part 8|2 pages

Part VIII: Constructibility Problems

chapter 38|7 pages

The Impossibility of Certain Constructions

chapter |1 pages

Section VIII in a Nutshell

part 9|2 pages

Part IX: Vector Spaces and Field Extensions

chapter 39|6 pages

Vector Spaces I

chapter 40|12 pages

Vector Spaces II

chapter 41|8 pages

Field Extensions and Kronecker’s Theorem

chapter 42|10 pages

Algebraic Field Extensions

chapter |3 pages

Section IX in a Nutshell

part 10|2 pages

Part X: Galois Theory

chapter 44|10 pages

The Splitting Field

chapter 45|6 pages

Finite Fields

chapter 46|12 pages

Galois Groups

chapter 47|12 pages

The Fundamental Theorem of Galois Theory

chapter 48|11 pages

Solving Polynomials by Radicals

chapter |3 pages

Section X in a Nutshell

chapter |26 pages

Hints and Solutions

chapter |5 pages

Guide to Notation