ABSTRACT

Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there

chapter 1|12 pages

The Natural Numbers

chapter 2|16 pages

The Integers

chapter 3|10 pages

Modular Arithmetic

chapter 4|14 pages

Polynomials with Rational Coefficients

chapter 5|14 pages

Factorization of Polynomials

chapter |2 pages

Section I in a Nutshell

chapter |2 pages

Section II. Rings, Domains, and Fields

chapter 6|16 pages

Rings

chapter 7|12 pages

Subrings and Unity

chapter 8|18 pages

Integral Domains and Fields

chapter 9|16 pages

Polynomials over a Field

chapter |4 pages

Section II in a Nutshell

chapter |2 pages

Section III. Unique Factorization

chapter 10|14 pages

Associates and Irreducibles

chapter 11|14 pages

Factorization and Ideals

chapter 12|8 pages

Principal Ideal Domains

chapter 13|12 pages

Primes and Unique Factorization

chapter 14|8 pages

Polynomials with Integer Coefficients

chapter 15|10 pages

Euclidean Domains

chapter |2 pages

Section III in a Nutshell

chapter 16|14 pages

Ring Homomorphisms

chapter 17|12 pages

The Kernel

chapter 18|10 pages

Rings of Cosets

chapter 19|12 pages

The Isomorphism Theorem for Rings

chapter 20|12 pages

Maximal and Prime Ideals

chapter 21|12 pages

The Chinese Remainder Theorem

chapter |2 pages

Section IV in a Nutshell

chapter |2 pages

Section V. Groups

chapter 22|12 pages

Symmetries of Figures in the Plane

chapter 23|14 pages

Symmetries of Figures in Space

chapter 24|16 pages

Abstract Groups

chapter 25|10 pages

Subgroups

chapter 26|14 pages

Cyclic Groups

chapter |2 pages

Section V in a Nutshell

chapter 27|10 pages

Group Homomorphisms

chapter 28|12 pages

Group Isomorphisms

chapter 29|10 pages

Permutations and Cayley’s Theorem

chapter 30|10 pages

More About Permutations

chapter 31|14 pages

Cosets and Lagrange’s Theorem

chapter 32|12 pages

Groups of Cosets

chapter 33|10 pages

The Isomorphism Theorem for Groups

chapter 34|14 pages

The Alternating Groups

chapter 36|6 pages

Solvable Groups

chapter |2 pages

Section VI in a Nutshell

chapter |2 pages

Section VII. Constructibility Problems

chapter 37|10 pages

Constructions with Compass and Straightedge

chapter 39|10 pages

The Impossibility of Certain Constructions

chapter |2 pages

Section VII in a Nutshell

chapter 40|8 pages

Vector Spaces I

chapter 41|16 pages

Vector Spaces II

chapter 42|10 pages

Field Extensions and Kronecker’s Theorem

chapter 43|14 pages

Algebraic Field Extensions

chapter |2 pages

Section VIII in a Nutshell

chapter |2 pages

Section IX Galois Theory

chapter 45|12 pages

The Splitting Field

chapter 46|8 pages

Finite Fields

chapter 47|14 pages

Galois Groups

chapter 48|16 pages

The Fundamental Theorem of Galois Theory

chapter 49|14 pages

Solving Polynomials by Radicals

chapter |4 pages

Section IX in a Nutshell

chapter |28 pages

Hints and Solutions

chapter |4 pages

Guide to Notation