- #1
- 22,183
- 3,324
- Author: Serge Lang
- Title: Algebra
- Amazon Link: https://www.amazon.com/dp/038795385X/?tag=pfamazon01-20
- Prerequisities: Good knowledge of undergrad Algebra
- Level: Grad
Table of Contents:
Code:
[LIST]
[*] The Basic Objects of Algebra
[LIST]
[*] Groups
[LIST]
[*] Monoids
[*] Groups
[*] Normal subgroups
[*] Cyclic groups
[*] Operations of a group on a set
[*] Sylow subgroups
[*] Direct sums and free abelian groups
[*] Finitely generated abelian groups
[*] The dual group
[*] Inverse limit and completion
[*] Categories and functors
[*] Free groups
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[*] Rings
[LIST]
[*] Rings and homomorphisms
[*] Commutative rings
[*] Polynomials and group rings
[*] Localization
[*] Principal and factorial rings
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[*] Modules
[LIST]
[*] Basic definitions
[*] The group of homomorphisms
[*] Direct products and sums of modules
[*] Free modules
[*] Vector spaces
[*] The dual space and dual module
[*] Modules over principal rings
[*] Euler-Poincare maps
[*] The snake lemma
[*] Direct and inverse limits
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[*] Polynomials
[LIST]
[*] Basic properties for polynomials in one variable
[*] Polynomials over a factorial ring
[*] Criteria for irreducibility
[*] Hilbert's theorem
[*] Partial fractions
[*] Symmetric polynomials
[*] Mason-Stothers theorem and the abc conjecture
[*] The resultant
[*] Power series
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[/LIST]
[*] Algebraic Equations
[LIST]
[*] Algebraic Extensions
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[*] Finite and algebraic extensions
[*] Algebraic closure
[*] Splitting fields and normal extensions
[*] Separable extensions
[*] Finite fields
[*] Inseparable extensions
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[*] Galois Theory
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[*] Galois extensions
[*] Examples and applications
[*] Roots of unity
[*] Linear independence of characters
[*] The norm and trace
[*] Cyclic extensions
[*] Solvable and radical extensions
[*] Abelian Kummer theory
[*] The equation X^n - a = 0
[*] Galois cohomology
[*] Non-abelian Kummer extensions
[*] Algebraic independence of homomorphisms
[*] The normal basis theorem
[*] Infinite Galois extensions
[*] The modular connection
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[*] Extensions of Rings
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[*] Integral ring extensions
[*] Integral Galois extensions
[*] Extension of homomorphisms
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[*] Transcendental Extensions
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[*] Transcendence bases
[*] Noether normalization theorem
[*] Linearly disjoint extensions
[*] Separable and regular extensions
[*] Derivations
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[*] Algebraic Spaces
[LIST]
[*] Hilbert's Nullstellensatz
[*] Algebraic sets, spaces and varieties
[*] Projections and elimination
[*] Resultant systems
[*] Spec of a ring
[/LIST]
[*] Noetherian Rings and Modules
[LIST]
[*] Basic criteria
[*] Associated primes
[*] Primary decomposition
[*] Nakayama's lemma
[*] Filtered and graded modules
[*] The Hilbert polynomial
[*] Indecomposable modules
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[*] Real Fields
[LIST]
[*] Ordered fields
[*] Real fields
[*] Real zeros and homomorphisms
[/LIST]
[*] Absolute Values
[LIST]
[*] Definitions, dependence, and independence
[*] Completions
[*] Finite extensions
[*] Valuations
[*] Completions and valuations
[*] Discrete valuations
[*] Zeros of polynomials in complete fields
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[/LIST]
[*] Linear Algebra and Representations
[LIST]
[*] Matrices and Linear Maps
[LIST]
[*] Matrices
[*] The rank of a matrix
[*] Matrices and linear maps
[*] Determinants
[*] Duality
[*] Matrices and bilinear forms
[*] Sesquilinear duality
[*] The simplicity of SL_2(F)/\pm 1
[*] The group SL_n(F),n\geq 3
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[*] Representation of One Endomorphism
[LIST]
[*] Representations
[*] Decomposition over one endomorphism
[*] The characteristic polynomial
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[*] Structure of Bilinear Forms
[LIST]
[*] Preliminaries, orthogonal sums
[*] Quadratic maps
[*] Symmetric forms, orthogonal bases
[*] Symmetric forms over ordered fields
[*] Hermitian forms
[*] The spectral theorem (hermitian case)
[*] The spectral theorem (symmetric case)
[*] Alternating forms
[*] The Pfaffian
[*] Witt's theorem
[*] The Witt group
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[*] The Tensor Product
[LIST]
[*] Tensor product
[*] Basic properties
[*] Flat modules
[*] Extension of the base
[*] Some functorial isomorphisms
[*] Tensor product of algebras
[*] The tensor algebra of a module
[*] Symmetric products
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[*] Semisimpliclty
[LIST]
[*] Matrices and linear maps over non-commutative rings
[*] Conditions defining semisimplicity
[*] The density theorem
[*] Semisimple rings
[*] Simple rings
[*] The Jacobson radical, base change, and tensor products
[*] Balanced modules
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[*] Representations of Finite Groups
[LIST]
[*] Representations and semisimplicity
[*] Characters
[*] 1-dimensional representations
[*] The space of class functions
[*] Orthogonality relations
[*] Induced characters
[*] Induced representations
[*] Positive decomposition of the regular character
[*] Supersolvable groups
[*] Brauer's theorem
[*] Field of definition of a representation
[*] Example: GL_2 over a finite field
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[*] The Alternating Product
[LIST]
[*] Definition and basic properties
[*] Fitting ideals
[*] Universal derivations and the de Rham complex
[*] The Clifford algebra
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[/LIST]
[*] Homological Algebra
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[*] General Homology Theory
[LIST]
[*] Complexes
[*] Homology sequence
[*] Euler characteristic and the Grothendieck group
[*] Injective modules
[*] Homotopies of morphisms of complexes
[*] Derived functors
[*] Delta-functors
[*] Bifunctors
[*] Spectral sequences
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[*] Finite Free Resolutions
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[*] Special complexes
[*] Finite free resolutions
[*] Unimodular polynomial vectors
[*] The Koszul complex
[/LIST]
[/LIST]
[*] Appendix: The Transcendence of e and \pi
[*] Appendix: Some Set Theory
[*] Bibliography
[*] Index
[/LIST]
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