目錄
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
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