1. Logic [1]
2. Sets and Classes [1]
3. Functions [3]
4. Relations and Partitions [6]
5. Products [7]
6. The Integers [9]
7. The Axiom of Choice, Order and Zorn’s Lemma [12]
8. Cardinal Numbers [15]
Chapter I: Groups [23]
1. Semigroups, Monoids and Groups [24]
2. Homomorphisms and Subgroups [30]
3. Cyclic Groups [35]
4. Cosets and Counting [37]
5. Normality, Quotient Groups, and Homomorphisms [41]
6. Symmetric, Alternating, and Dihedral Groups [46]
7. Categories: Products, Coproducts, and Free Objects [52]
8. Direct Products and Direct Sums [59]
9. Free Groups, Free Products, Generators & Relations [64]
Chapter II: The Structure of Groups [70]
1. Free Abelian Groups [70]
2. Finitely Generated Abelian Groups [76]
3. The Krull-Schmidt Theorem [83]
4. The Action of a Group on a Set [88]
5. The Sylow Theorems [92]
6. Classification of Finite Groups [96]
7. Nilpotent and Solvable Groups [100]
8. Normal and Subnormal Series [107]
Chapter III: Rings [114]
1. Rings and Homomorphisms [115]
2. Ideals [122]
3. Factorization in Commutative Rings [135]
4. Rings of Quotients and Localization [142]
5. Rings of Polynomials and Formal Power Series [149]
6. Factorization in Polynomial Rings [157]
Chapter IV: Mod u les [114]
1. Modules, Homomorphisms and Exact Sequences [169]
2. Free Modules and Vector Spaces [180]
3. Projective and Injective Modules [190]
4. Hom and Duality [199]
5. Tensor Products [207]
6. Modules over a Principal Ideal Domain [218]
7. Algebras [226]
Chapter V: Fields and Galois Theory [168]
1. Field Extensions [231]
Appendix: Ruler and Compass Constructions [238]
2. The Fundamental Theorem [243]
Appendix: Symmetric Rational Functions [252]
3. Splitting Fields, Algebraic Closure and Normality [257]
Appendix: The Fundamental Theorem of Algebra [265]
4. The Galois Group of a Polynomial [269]
5. Finite Fields [278]
6. Separability. [282]
7. Cyclic Extensions [289]
8. Cyclotomic Extensions [297]
9. Radical Extensions [302]
Appendix: The General Equation of Degree [307]
Chapter VI: The Structure of Fields [311]
1. Transcendence Bases [311]
2. Linear Disjointness and Separability [318]
Chapter VII: Linear Algebra [327]
1. Matrices and Maps [328]
2. Rank and Equivalence [335]
Appendix: Abelian Groups Defined by
Generators and Relations [343]
3. Determinants [348]
4. Decomposition of a Single Linear Transformation and Similarity [355]
5. The Characteristic Polynomial, Eigenvectors and Eigenvalues [366]
Chapter VIII: Commutative Ringsand Modules [371]
1. Chain Conditions [372]
2. Prime and Primary Ideals [377]
3. Primary Decomposition [383]
4. Noetherian Rings and Modules [387]
5. Ring Extensions [394]
6. Dedekind Domains [400]
7. The Hilbert Nullstellensatz [409]
Chapter IX: The Structure of Rings [414]
1. Simple and Primitive Rings [415]
2. The Jacobson Radical [424]
3. Semisimple Rings [434]
4. The Prime Radical; Prime and Semiprime Rings [444]
5. Algebras [450]
6. Division Algebras [456]
Chapter X: Categories [464]
1. Functors and Natural Transformations [465]
2. Adjoint Functors [476]
3. Morphisms [480]
List of Symbols [485]
