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Andrew O. Lindstrum, "Abstract Algebra (Holden-Day series in mathematics)"

Table of Contents

1. Sets, Mappings, Laws of Composition, and the Natural Numbers

1.1 Logic

1.2 Sets

1.3 Mapping of one set into another set

1.4 Set products and laws of composition

1.5 Properties of laws of internal composition

1.6 Natural numbers

1.7 Addition of natural numbers

1.8 The cancellation law

1.9 Multiplication of natural numbers

1.10 Relations

1.11 Inequality in N

2. Semigroups, Equivalence Relations and the Rational Integers

2.1 Semigroups

2.2 Products of mappings

2.3 The associative law generalized

2.4 Subsemigroups

2.5 Neutral elements and inverse elements

2.6 Definition of a group

2.7 A theorem about mappings

2.8 Equivalence relations

2.9 Semigroup products of semigroups

2.10 Composition table of a semigroup

2.11 Homomorphisms and isomorphisms

2.12 Inducing laws of composition in quotient sets

2.13 Definition of the rational integers

2.14 Absolute value of rational integers

2.15 Exponents 37

2.16 Divisibility in a semigroup 37

2.17 Divisibility in Z 39

2.18 Unique factorization 41

2.19 Congruences 42

3. Groups 46

3.1 General properties of subgroups 46

3.2 Cyclic groups and subgroups 49

3.3 Equivalence relations in a group 52

3.4 Homomorphisms and isomorphisms of groups 57

3.5 Two families of groups 62

3.6 Conjugates 63

3.7 Direct products 65

3.8 Products of subgroups of groups 68

3.9 Free groups 70

3.10 Sylow theorems 74

3.11 Permutations and permutation groups 78

3.12 Finite abelian groups 83

3.13 Automorphisms and endomorphisms of the four-group, D4 87

3.14 Composition series 89

4. Systems with more than one Law of Composition 92

4.1 Rings, fields, integral domains 92

4.2 Laws of external composition and groups with operators 97

4.3 Algebraic systems and homomorphisms 100

4.4 Modules 102

4.5 Linear dependence in an R-module 105

4.6 Vector spaces 108

4.7 Modules of linear combinations and linear relations 110

4.8 Algebras 111

4.9 Quarternions 114

5. Polynomials, Factorization, Ideals, and Extension of Fields

5.1 Polynomials

5.2 Polynomials and polynomial functions

5.3 Gaussian semigroups and Gaussian domains

5.4 Euclidean domains

5.5 Polynomials in two indeterminates

5.6 Fields of quotients of polynomials

5.7 Ideals

5.8 Principal ideal rings

5.9 Quotient rings and equivalence relations in a ring

5.10 Prime and maximal ideals

5.11 Extension of fields

5.12 Structure of fields

5.13 Adjunction of several elements to a field

5.14 Trisection of an arbitrary angle

5.15 Extensions of isomorphisms

6. Fields 143

6.1 Prime fields 143

6.2 Conjugate elements and automorphisms of fields 144

6.3 Normal extensions of fields and normal polynomials 146

6.4 Separability 148

6.5 Subfields and automorphisms 150

6.6 Roots of unity 152

6.7 Finite fields 153

6.8 Primitive elements 155

6.9 The Galois theory of fields 156

6.10 The cyclotomic field 158

6.11 Pure extension fields 160

6.12 Solvability by radicals 162

7. Linear Mappings and Matrices 165

7.1 Linear mappings of modules 165

7.2 Matrices 167

7.3 Rank 170

7.4 Change of basis 1 72

7.5 Coordinates 173

7.6 Application to linear equations 175

7.7 Row-equivalence and elementary operations 176

7.8 A particular kind of basis for a vector subspace 178

7.9 Equivalence of matrices over a field

7.10 Equivalence of matrices over a Euclidean ring

7.11 Equivalence over F : similarity

7.12 Vector subspaces invariant under a linear transformation

7.13 Minimum polynomials

7.14 Cyclic spaces and transformations

7.15 Noncyclic linear transformations

7.16 Invariant factors and sini.i!aity invariants

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