导语

  

内容提要

  

    本书是一部很有影响力的研究生教材，全面介绍了代数的基本概念。本书的突出特点是书中不但保留了代数的经典内容，同时也介绍了从 范畴理论和同调代数思考的学习方式，各章有大量习题。本书可做为研究生教材，学时一年。
    目次：（一）代数基本内容：群；环；模；多项式。（二）代数方程：代数 扩张；伽罗瓦理论；环的扩张；超越扩张；代数空间；诺特环和模；实域；绝对值。（三）线性代数和表示：矩阵映射和线性映射；单同态表示；双线性型结构；张量积；半单性；有限群表示；交错积。（四）一般同调理论；有限自由解。    
    读者对象：数学专业的研究生及相关专业的研究人员。

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