Language: English
ISBN/ISSN: 9787030408778
2014-06; Soft Cover
Preface
Chapter 1 Groups 1
1.1 Semigroups, monoids and groups 1
1.2 Subgroups 6
1.3 Theactionofagrouponaset 11
1.4 The Sylow theorem 18
1.5 Homomorphisms and normal subgroups 20
1.6 Directproductsanddirectsums 29
1.7 Simple groups 36
1.8 Nilpotent groups and solvable groups 39
Chapter 2 Modules 44
2.1 Rings and ring homomorphisms 44
2.2 Modules and free modules 55
2.3 Projectivemodulesandinjectivemodules 67
2.4 Homologicaldimensionsandsemisimplerings 75
2.5 Tensor product and weak dimension 83
2.6 Localization 95
2.7 Noetherianmodules and UFD 104
2.8 Finitely generated modules over a PID 115
Chapter 3 Fields and Galois Theory of Equations 129
3.1 Extensionsof .elds 129
3.2 Splitting .elds, and normality 137
3.3 ThemaintheoremofGaloistheory 147
3.4 Radical extensions 155
3.5 Constructionwithstraight-edgeandcompass 157
3.6 The Hilbert Nullstellensatz 160
Chapter 4 Introduction of Various Algebras 167
4.1 Associativealgebras 167
4.2 CoassociativecoalgebrasandHopfalgebras 178
4.3 Nonassociative algebras 182
Chapter 5 Category 193
5.1 Category: Direct limits and colimits 193
5.2 Functors and natural transformations 198
5.3 Abelian categories and homological groups 207
Bibliography 217
Index 219
Preface
Chapter 1 Groups 1
1.1 Semigroups, monoids and groups 1
1.2 Subgroups 6
1.3 Theactionofagrouponaset 11
1.4 The Sylow theorem 18
1.5 Homomorphisms and normal subgroups 20
1.6 Directproductsanddirectsums 29
1.7 Simple groups 36
1.8 Nilpotent groups and solvable groups 39
Chapter 2 Modules 44
2.1 Rings and ring homomorphisms 44
2.2 Modules and free modules 55
2.3 Projectivemodulesandinjectivemodules 67
2.4 Homologicaldimensionsandsemisimplerings 75
2.5 Tensor product and weak dimension 83
2.6 Localization 95
2.7 Noetherianmodules and UFD 104
2.8 Finitely generated modules over a PID 115
Chapter 3 Fields and Galois Theory of Equations 129
3.1 Extensionsof .elds 129
3.2 Splitting .elds, and normality 137
3.3 ThemaintheoremofGaloistheory 147
3.4 Radical extensions 155
3.5 Constructionwithstraight-edgeandcompass 157
3.6 The Hilbert Nullstellensatz 160
Chapter 4 Introduction of Various Algebras 167
4.1 Associativealgebras 167
4.2 CoassociativecoalgebrasandHopfalgebras 178
4.3 Nonassociative algebras 182
Chapter 5 Category 193
5.1 Category: Direct limits and colimits 193
5.2 Functors and natural transformations 198
5.3 Abelian categories and homological groups 207
Bibliography 217
Index 219
