Algebraic Structures [1 ed.] 0201041731, 9780201041736

Table of contents :
Contents

CHAPTER I

The Integers
1. Terminology of sets . . . . . . . . . . . . . . . . . . 1
2. Basic properties . . . . . . . . . . . . . . . . . . . . 2
3. Greatest common divisor . . . . . . . . . . . . . . . . 5
4. Unique factorization . . . . . . . . . . . . . . . . . . 6
5. Equivalence relations and congruences. . . . . . . . . . 8

CHAPTER II

Groups
1. Groups and examples . . . . . . . . . . . . . . 12
2. Mappings . . . . . . . . . . . . . . . . . . . . . 17
3. Homomorphisms . . . . . . . . . . . . . . . . . . . . . 21
4. Cosets and normal subgroups . . . . . . . . . . . . . . 26
5. Permutation groups. . . . . . . . . . . . . . . . . . . 32
6. Cyclic groups . . . . . . . . . . . . . . . . . . . . . 39

CHAPTER III

Rings
1. Rings . . . . . . . . . . . . . . . . . . . 43
2. Ideals . . . . . . . . . . . . . . . . . . . 46
3. Homomorphisms . . . . . . . . . . . . . . . . 48
4. Quotient fields . . . . . . . . . . . . . . . . 54

CHAPTER IV

Polynomials
1. Euclidean algorithm . . . . . . . . . . . . . . . 58
2. Greatest common divisor . . . . . . . . . . . . . 63
3. Unique factorization . . . . . . . . . . . . . . 65
4. Partial fractions . . . . . . . . . . . . . . . . 70
5. Polynomials over the integers . . . . . . . . . . 76
6. Transcendental elements . . . . . . . . . . . . . 79
7. Polynomials in several variables . . . . . . . . 84


CHAPTER V

Vector Spaces and Modules
1. Vector spaces and bases . . . . . . . . . . . . . . 86
2. Dimension of a vector space . . . . . . . . . . . . 92
3. Modules . . . . . . . . . . . . . . . . . . . . . . 94

CHAPTER VI

Field Theory
1. Algebraic extensions . . . . . . . . . . . . . . . 102
2. Embeddings . . . . . . . . . . . . . . . . . . . . 105
3. Splitting fields. . . . . . . . . . . . . . . . . . 110
4. Fundamental theorem. . . . . . . . . . . . . . . . 111
5. Quadratic and cubic extensions. . . . . . . . . . 113
6. Solvability by radicals . . . . . . . . . . . . . 115
7. Infinite extensions . . . . . . . . . . . . . . . . 118

CHAPTER VII

The Real and Complex Numbers
1. Ordering of rings. . . . . . . . . . . . . . . . 120
2. Preliminaries. . . . . . . . . . . . . . . . . . 123
3. Construction of the real numbers . . . . . . . . 126
4. Decimal expansions . . . . . . . . . . . . . . . 133
5. The complex numbers. . . . . . . . . . . . . . . 136

CHAPTER VIII

Sets
1. More terminology . . . . . . . . . . . . . . . 141
2. Zorn’s lemma . . . . . . . . . . . . . . . . . 144
3. Cardinal numbers . . . . . . . . . . . . . . . 148
4. Well-ordering . . . . . . . . . . . . . . . . . 158
5. Proof of Zorn’s lemma. . . . . . . . . . . . . 160

Appendix

1. The natural numbers . . . . . . . . . . . . . . 164
2. The integers . . . . . . . . . . . . . . . . . 168

Index . . . . . . . . . . . . . . . . . . . 171