Pub. Date:
Springer New York
Algebra / Edition 3

Algebra / Edition 3

by Serge LangSerge Lang
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"Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books." NOTICES OF THE AMS

"The author has an impressive knack for presenting the important and interesting ideas of algebra in just the right way, and he never gets bogged down in the dry formalism which pervades some parts of algebra." MATHEMATICAL REVIEWS

This book is intended as a basic text for a one-year course in algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text.

Product Details

ISBN-13: 9780387953854
Publisher: Springer New York
Publication date: 06/21/2005
Series: Graduate Texts in Mathematics , #211
Edition description: 3rd rev. ed. 2002. Corr. 4th printing 2005
Pages: 918
Product dimensions: 6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

Part 1The Basic Objects of Algebra
Chapter IGroups3
3.Normal subgroups13
4.Cyclic groups23
5.Operations of a group on a set25
6.Sylow subgroups33
7.Direct sums and free abelian groups36
8.Finitely generated abelian groups42
9.The dual group46
10.Inverse limit and completion49
11.Categories and functors53
12.Free groups66
Chapter IIRings83
1.Rings and homomorphisms83
2.Commutative rings92
3.Polynomials and group rings97
5.Principal and factorial rings111
Chapter IIIModules117
1.Basic definitions117
2.The group of homomorphisms122
3.Direct products and sums of modules127
4.Free modules135
5.Vector spaces139
6.The dual space and dual module142
7.Modules over principal rings146
8.Euler-Poincare maps155
9.The snake lemma157
10.Direct and inverse limits159
Chapter IVPolynomials173
1.Basic properties for polynomials in one variable173
2.Polynomials over a factorial ring180
3.Criteria for irreducibility183
4.Hilbert's theorem186
5.Partial fractions187
6.Symmetric polynomials190
7.Mason-Stothers theorem and the abc conjecture194
8.The resultant199
9.Power series205
Part 2Algebraic Equations
Chapter VAlgebraic Extensions223
1.Finite and algebraic extensions225
2.Algebraic closure229
3.Splitting fields and normal extensions236
4.Separable extensions239
5.Finite fields244
6.Inseparable extensions247
Chapter VIGalois Theory261
1.Galois extensions261
2.Examples and applications269
3.Roots of unity276
4.Linear independence of characters282
5.The norm and trace284
6.Cyclic extensions288
7.Solvable and radical extensions291
8.Abelian Kummer theory293
9.The equation X[superscript n] - a = 0297
10.Galois cohomology302
11.Non-abelian Kummer extensions304
12.Algebraic independence of homomorphisms308
13.The normal basis theorem312
14.Infinite Galois extensions313
15.The modular connection315
Chapter VIIExtensions of Rings333
1.Integral ring extensions333
2.Integral Galois extensions340
3.Extension of homomorphisms346
Chapter VIIITranscendental Extensions355
1.Transcendence bases355
2.Noether normalization theorem357
3.Linearly disjoint extensions360
4.Separable and regular extensions363
Chapter IXAlgebraic Spaces377
1.Hilbert's Nullstellensatz377
2.Algebraic sets, spaces and varieties381
3.Projections and elimination388
4.Resultant systems401
5.Spec of a ring405
Chapter XNoetherian Rings and Modules413
1.Basic criteria413
2.Associated primes416
3.Primary decomposition421
4.Nakayama's lemma424
5.Filtered and graded modules426
6.The Hilbert polynomial431
7.Indecomposable modules439
Chapter XIReal Fields449
1.Ordered fields449
2.Real fields451
3.Real zeros and homomorphisms457
Chapter XIIAbsolute Values465
1.Definitions, dependence, and independence465
3.Finite extensions476
5.Completions and valuations486
6.Discrete valuations487
7.Zeros of polynomials in complete fields491
Part 3Linear Algebra and Representations
Chapter XIIIMatrices and Linear Maps503
2.The rank of a matrix506
3.Matrices and linear maps507
6.Matrices and bilinear forms527
7.Sesquilinear duality531
8.The simplicity of SL[subscript 2](F)/[plus or minus]1536
9.The group SL[subscript n](F), n [greater than or equal] 3540
Chapter XIVRepresentation of One Endomorphism553
2.Decomposition over one endomorphism556
3.The characteristic polynomial561
Chapter XVStructure of Bilinear Forms571
1.Preliminaries, orthogonal sums571
2.Quadratic maps574
3.Symmetric forms, orthogonal bases575
4.Symmetric forms over ordered fields577
5.Hermitian forms579
6.The spectral theorem (hermitian case)581
7.The spectral theorem (symmetric case)584
8.Alternating forms586
9.The Pfaffian588
10.Witt's theorem589
11.The Witt group594
Chapter XVIThe Tensor Product601
1.Tensor product601
2.Basic properties607
3.Flat modules612
4.Extension of the base623
5.Some functorial isomorphisms625
6.Tensor product of algebras629
7.The tensor algebra of a module632
8.Symmetric products635
Chapter XVIISemisimplicity641
1.Matrices and linear maps over non-commutative rings641
2.Conditions defining semisimplicity645
3.The density theorem646
4.Semisimple rings651
5.Simple rings654
6.The Jacobson radical, base change, and tensor products657
7.Balanced modules660
Chapter XVIIIRepresentations of Finite Groups663
1.Representations and semisimplicity663
3.1-dimensional representations671
4.The space of class functions673
5.Orthogonality relations677
6.Induced characters686
7.Induced representations688
8.Positive decomposition of the regular character699
9.Supersolvable groups702
10.Brauer's theorem704
11.Field of definition of a representation710
12.Example: GL[subscript 2] over a finite field712
Chapter XIXThe Alternating Product731
1.Definition and basic properties731
2.Fitting ideals738
3.Universal derivations and the de Rham complex746
4.The Clifford algebra749
Part 4Homological Algebra
Chapter XXGeneral Homology Theory761
2.Homology sequence767
3.Euler characteristic and the Grothendieck group769
4.Injective modules782
5.Homotopies of morphisms of complexes787
6.Derived functors790
9.Spectral sequences814
Chapter XXIFinite Free Resolutions835
1.Special complexes835
2.Finite free resolutions839
3.Unimodular polynomial vectors846
4.The Koszul complex850
Appendix 1The Transcendence of e and [Pi]867
Appendix 2Some Set Theory875

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