Algebra /
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| Author / Creator: | Lang, Serge, 1927-2005. |
|---|---|
| Edition: | Rev. 3rd ed. |
| Imprint: | New York : Springer, ©2002. |
| Description: | xv, 914 pages ; 24 cm. |
| Language: | English |
| Series: | Graduate texts in mathematics ; 211 Graduate texts in mathematics ; 211. |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4636669 |
Table of Contents:
- Part 1. 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 abc conjecture
- 8.. The resultant
- 9.. Power series
- Part 2. 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[superscript n] - a = 0
- 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. Noetherian 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 3. 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 SL[subscript 2](F)/[plus or minus]1
- 9.. The group SL[subscript n](F), n [greater than or equal] 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. Semisimplicity
- 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.. 1-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: GL[subscript 2] 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 4. 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 [Pi]
- Appendix 2. Some Set Theory
- Bibliography
- Index