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Kan Onn Kit

A Complete Introduction to Abstract Algebra

A full first course in abstract algebra: groups, subgroups, homomorphisms and quotients through the Sylow theorems, then rings, ideals and factorization, and fields through Galois theory and the fundamental theorem of algebra. Every proof step is spelled out rather than left as an exercise for the reader.

A Complete Introduction to Abstract Algebra

432 exercises36 sections
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Sections
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Preliminaries

68
Sets7
Mathematical Logic7
Proof Writing10
Algebra12
Relations and Functions14
Elementary Number Theory18

Groups

146
Introduction To Groups3
Basics of Groups10
Subgroups19
Homomorphisms and Isomorphisms19
Symmetry Groups6
Direct Products of Groups9
Further Properties of Homomorphisms11
More Types of Groups16
Group Actions17
Sylow Theorems17
Abelian Groups8
Composition Series6
Simple Groups5

Rings

127
Introduction to Rings1
Basics of Rings13
Integral Domains6
Ideals and Quotient Rings20
Ring Homomorphisms and Isomorphisms21
Polynomial Rings21
Factorization of Polynomials16
Domains and Factorization23
Rings and Encryption6

Fields

69
Basics of Fields11
Vector Spaces14
Extension Fields and Splitting Fields14
Algebraic Extensions15
Finite Fields12
Geometric Constructions3

Galois

22
Galois Theory19
The Fundamental Theorem of Algebra3

Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.