Preface
Chapter 1 Groups
1.1 Semigroups, monoids and groups
1.2 Subgroups
1.3 The action of a group on a set
1.4 The Sylow theorem
1.5 Homomorphisms and normal subgroups
1.6 Direct products and direct sums
1.7 Simple groups
1.8 Nilpotent groups and solvable groups
Chapter 2 Modules
2.1 Rings and ring homomorphisms
2.2 Modules and free modules
2.3 Projective modules and injective modules
2.4 Homological dimensions and semisimple rings
2.5 Tensor product and weak dimension
2.6 Localization
2.7 Noetherian modules and UFD
2.8 Finitely generated modules over a PID
Chapter 3 Fields and Galois Theory of Equations
3.1 Extensions of fields
3.2 Splitting fields, and normality
3.3 The main theorem of Galois theory
3.4 Radical extensions
3.5 Construction with straight-edge and compass
3.6 The Hilbert Nullstellensatz
Chapter 4 Introduction of Various Algebras
4.1 Associative algebras
4.2 Coassociative coalgebras and Hopf algebras
4.3 Nonassociative algebras
Chapter 5 Category
5.1 Category: Direct limits and colimits
5.2 Functors and natural transformations
5.3 Abelian categories and homological groups
Bibliography
Index
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