College Algebra
Introduction to Modern Algebra
David Joyce
Clark University
Version 0.0.6, 3 Oct 2008
1 Copyright
(C) 2008.
1
ii
I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write this book. I’m sorry that he did not live to see it finished.
Contents
1 Introduction
1.1 Structures in Modern Algebra . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.5 Other algebraic structures besides fields, rings, and groups . . . . .
1.2 Isomorphisms, homomorphisms, etc. . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . .
1.2.4 Endomorphisms and automorphisms . . . . . . . . . . . . . . . . .
1.3 A little number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Mathematical induction on the natural numbers N . . . . . . . . .
1.3.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . .
1.4 The fundamental theorem of arithmetic: the unique factorization theorem .
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2 Fields
2.1 Introduction to fields . . . . . . . . . . . . . . . . . . .
2.1.1 Definition of fields . . . . . . . . . . . . . . . .
2.1.2 Subtraction, division,
David Joyce
Clark University
Version 0.0.6, 3 Oct 2008
1 Copyright
(C) 2008.
1
ii
I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write this book. I’m sorry that he did not live to see it finished.
Contents
1 Introduction
1.1 Structures in Modern Algebra . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Operations on sets . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.5 Other algebraic structures besides fields, rings, and groups . . . . .
1.2 Isomorphisms, homomorphisms, etc. . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . .
1.2.4 Endomorphisms and automorphisms . . . . . . . . . . . . . . . . .
1.3 A little number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Mathematical induction on the natural numbers N . . . . . . . . .
1.3.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . .
1.4 The fundamental theorem of arithmetic: the unique factorization theorem .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
1
3
3
5
7
7
8
9
10
11
11
12
12
14
15
17
2 Fields
2.1 Introduction to fields . . . . . . . . . . . . . . . . . . .
2.1.1 Definition of fields . . . . . . . . . . . . . . . .
2.1.2 Subtraction, division,