目录 Part One The Basic Objects of Algebra Chapter I Groups 1. Monoids 2. Groups 3. Normal subgroups 4. Cyclic groups 5. Operations of a group on a set 6. Sylow subgroups 7. Direct sums and free abelian groups 8. Finitely generated abelian groups 9. The dual group 10. Inverse limit and completion 11. Categories and functors 12. Free groups Chapter II Rings 1. Rings and homomorphisms 2. Commutative rings 3. Polynomials and group rings 4. Localization 5. Principal and factorial rings Chapter III Modules 1. Basic definitions 2. The group of homomorphisms 3. Direct products and sums of modules 4. Free modules 5. Vector spaces 6. The dual space and dual module 7. Modules over principal rings 8. Euler-Poincare maps 9. The snake lemma 10. Direct and inverse limits Chapter IV Polynomials 1. Basic properties for polynomials in one variable 2. Polynomials over a factorial ring 3. Criteria for irreducibility 4. Hilbert's theorem 5. Partial fractions 6. Symmetric polynomials 7. Mason-Stothers theorem and the abe conjecture 8. The resultant 9. Power series
Part Two Algebraic Equations Chapter V Algebraic Extensions 1. Finite and algebraic extensions 2. Algebraic closure 3. Splitting fields and normal extensions 4. Separable extensions 5. Finite fields 6. Inseparable extensions Chapter VI Galois Theory 1. Galois extensions 2. Examples and applications 3. Roots of unity 4. Linear independence of characters 5. The norm and trace 6. Cyclic extensions 7. Solvable and radical extensions 8. Abelian Kummer theory 9. The equation X" - a = 10. Galois cohomology 11. Non-abelian Kummer extensions 12. Algebraic independence of homomorphisms 13. The normal basis theorem 14. Infinite Galois extensions 15. The modular connection Chapter VII Extensions of Rings 1. Integral ring extensions 2. Integral Galois extensions 3. Extension of homomorphisms Chapter VIII Transcendental Extensions 1. Transcendence bases 2. Noether normalization theorem 3. Linearly disjoint extensions 4. Separable and regular extensions 5. Derivations Chapter IX Algebraic Spaces 1. Hilbert's Nullstellensatz 2. Algebraic sets, spaces and varieties 3. Projections and elimination 4. Resultant systems 5. Spec of a ring Chapter X Noetherlan Rings and Modules 1. Basic criteria 2. Associated primes 3. Primary decomposition 4. Nakayama's lemma 5. Filtered and graded modules 6. The Hilbert polynomial 7. Indecomposable modules Chapter XI Real Fields 1. Ordered fields 2. Real fields 3. Real zeros and homomorphisms Chapter XII Absolute Values 1. Definitions, dependence, and independence 2. Completions 3. Finite extensions 4. Valuations 5. Completions and valuations 6. Discrete valuations 7. Zeros of polynomials in complete fields
Part Three Linear Algebra and Representations Chapter XIII Matrices and Linear Maps 1. Matrices 2. The rank of a matrix 3. Matrices and linear maps 4. Determinants 5. Duality 6. Matrices and bilinear forms 7. Sesquilinear duality 8. The simplicity of SL2(F)/±1 9. The group SLn(F), n ≥3 Chapter XIV Representation of One Endomorphism 1. Representations 2. Decomposition over one endomorphism 3. The characteristic polynomial Chapter XV Structure of Bilinear Forms 1. Preliminaries, orthogonal sums 2. Quadratic maps 3. Symmetric forms, orthogonal bases 4. Symmetric forms over ordered fields 5. Hermitian forms 6. The spectral theorem (hermitian case) 7. The spectral theorem (symmetric case) 8. Alternating forms 9. The Pfaffian 10. Witt's theorem 11. The Witt group Chapter XVI The Tensor Product 1. Tensor product 2. Basic properties 3. Flat modules 4. Extension of the base 5. Some functorial isomorphisms 6. Tensor product of algebras 7. The tensor algebra of a module 8. Symmetric products Chapter XVII Semisimpllcity 1. Matrices and linear maps over non-commutative rings 2. Conditions defining semisimplicity 3. The density theorem 4. Semisimple rings 5. Simple rings 6. The Jacobson radical, base change, and tensor products 7. Balanced modules Chapter XVIII Representations of Finite Groups 1. Representations and semisimplicity 2. Characters 3. l-dimensional representations 4. The space of class functions 5. Orthogonality relations 6. Induced characters 7. Induced representations 8. Positive decomposition of the regular character 9. Supersolvable groups 10. Brauer's theorem 11. Field of definition of a representation 12. Example: GL2 over a finite field Chapter XIX The Alternating Product 1. Definition and basic properties 2. Fitting ideals 3. Universal derivations and the de Rham complex 4. The Clifford algebra
Part Four Homological Algebra Chapter XX General Homology Theory 1. Complexes 2. Homology sequence 3. Euler characteristic and the Grothendieck group 4. Injective modules 5. Homotopies of morphisms of complexes 6. Derived functors 7. Delta-functors 8. Bifunctors 9. Spectral sequences Chapter XXI Finite Free Resolutions 1. Special complexes 2. Finite free resolutions 3. Unimodular polynomial vectors 4. The Koszul complex Appendix 1 The Transcendence of e and π Appendix 2 Some Set Theory Bibliography Index
