Abstract Algebra

By: Pierre Antoine Grillet

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Hardcover | 24 July 2007 | Edition Number 2

At a Glance

Hardcover
688 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 3.66

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About the first edition:

"The text is geared to the needs of the beginning graduate student, covering with complete, well-written proofs the usual major branches of groups, rings, fields, and modules...[n]one of the material one expects in a book like this is missing, and the level of detail is appropriate for its intended audience." (Alberto Delgado, MathSciNet)

"This text promotes the conceptual understanding of algebra as a whole, and that with great methodological mastery. Although the presentation is predominantly abstract...it nevertheless features a careful selection of important examples, together with a remarkably detailed and strategically skillful elaboration of the more sophisticated, abstract theories." (Werner Kleinert, Zentralblatt)

For the new edition, the author has completely rewritten the text, reorganized many of the sections, and even cut or shortened material which is no longer essential. He has added a chapter on Ext and Tor, as well as a bit of topology.

Industry Reviews

From the reviews of the second edition:

"This book certainly has more than enough material for a two-semester algebra course for first-year graduate students. ... The balance among various topics will please some instructors more than others." (Miklos Bona, MathDL, September, 2007)

"The book under review is the second edition of the author's sweeping text originally titled 'Algebra' ... . All together, the second edition of P. A. Grillet's standard primer is a masterly, clearly written, comprehensive and self-contained algebra text for graduate students, with a plentiful supply of additional advanced material ... . Moreover, the book under review, in its didactically oriented disposition, represents an excellent source for teachers and instructors of graduate courses in abstract algebra as well." (Werner Kleinert, Zentralblatt MATH, Vol. 1122 (24), 2007)

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Preface	p. vii
Groups	p. 1
Semigroups	p. 1
Groups	p. 8
Subgroups	p. 12
Homomorphisms	p. 18
The Isomorphism Theorems	p. 23
Free Groups	p. 27
Presentations	p. 31
Free Products	p. 37
Structure of Groups	p. 43
Direct Products	p. 43
The Krull-Schmidt Theorem	p. 48
Group Actions	p. 54
Symmetric Groups	p. 58
The Sylow Theorems	p. 64
Small Groups	p. 67
Composition Series	p. 70
The General Linear Group	p. 76
Solvable Groups	p. 83
Nilpotent Groups	p. 89
Semidirect Products	p. 92
Group Extensions	p. 95
Rings	p. 105
Rings	p. 105
Subrings and Ideals	p. 109
Homomorphisms	p. 112
Domains and Fields	p. 116
Polynomials in One Variable	p. 119
Polynomials in Several Variables	p. 125
Formal Power Series	p. 130
Principal Ideal Domains	p. 133
Rational Fractions	p. 139
Unique Factorization Domains	p. 141
Noetherian Rings	p. 146
Grobner Bases	p. 148
Field Extensions	p. 155
Fields	p. 155
Extensions	p. 159
Algebraic Extensions	p. 164
The Algebraic Closure	p. 165
Separable Extensions	p. 169
Purely Inseparable Extensions	p. 173
Resultants and Discriminants	p. 176
Transcendental Extensions	p. 181
Separability	p. 184
Galois Theory	p. 191
Splitting Fields	p. 191
Normal Extensions	p. 193
Galois Extensions	p. 197
Infinite Galois Extensions	p. 200
Polynomials	p. 204
Cyclotomy	p. 211
Norm and Trace	p. 215
Solvability by Radicals	p. 221
Geometric Constructions	p. 226
Fields with Orders or Valuations	p. 231
Ordered Fields	p. 231
Real Fields	p. 234
Absolute Values	p. 239
Completions	p. 243
Extensions	p. 247
Valuations	p. 251
Extending Valuations	p. 256
Hensel's Lemma	p. 261
Filtrations and Completions	p. 266
Commutative Rings	p. 273
Primary Decomposition	p. 273
Ring Extensions	p. 277
Integral Extensions	p. 280
Localization	p. 285
Dedekind Domains	p. 290
Algebraic Integers	p. 294
Galois Groups	p. 297
Minimal Prime Ideals	p. 300
Krull Dimension	p. 304
Algebraic Sets	p. 307
Regular Mappings	p. 310
Modules	p. 315
Definition	p. 315
Homomorphisms	p. 320
Direct Sums and Products	p. 324
Free Modules	p. 329
Vector Spaces	p. 334
Modules over Principal Ideal Domains	p. 336
Jordan Form of Matrices	p. 342
Chain Conditions	p. 346
Grobner Bases	p. 350
Semisimple Rings and Modules	p. 359
Simple Rings and Modules	p. 359
Semisimple Modules	p. 362
The Artin-Wedderburn Theorem	p. 366
Primitive Rings	p. 370
The Jacobson Radical	p. 374
Artinian Rings	p. 377
Representations of Groups	p. 380
Characters	p. 386
Complex Characters	p. 389
Projectives and Injectives	p. 393
Exact Sequences	p. 393
Pullbacks and Pushouts	p. 397
Projective Modules	p. 401
Injective Modules	p. 403
The Injective Hull	p. 408
Hereditary Rings	p. 411
Constructions	p. 415
Groups of Homomorphisms	p. 415
Properties of Hom	p. 419
Direct Limits	p. 423
Inverse Limits	p. 429
Tensor Products	p. 434
Properties of Tensor Products	p. 441
Dual Modules	p. 448
Flat Modules	p. 450
Completions	p. 456
Ext and Tor	p. 463
Complexes	p. 463
Resolutions	p. 471
Derived Functors	p. 478
Ext	p. 487
Tor	p. 493
Universal Coefficient Theorems	p. 497
Cohomology of Groups	p. 500
Projective Dimension	p. 507
Global Dimension	p. 510
Algebras	p. 515
Algebras over a Ring	p. 515
The Tensor Algebra	p. 518
The Symmetric Algebra	p. 521
The Exterior Algebra	p. 523
Tensor Products of Algebras	p. 527
Tensor Products of Fields	p. 530
Simple Algebras over a Field	p. 534
Lattices	p. 539
Definitions	p. 539
Complete Lattices	p. 543
Modular Lattices	p. 545
Distributive Lattices	p. 549
Boolean Lattices	p. 553
Universal Algebra	p. 559
Universal Algebras	p. 559
Word Algebras	p. 564
Varieties	p. 567
Subdirect Products	p. 574
Categories	p. 581
Definition and Examples	p. 581
Functors	p. 586
Limits and Colimits	p. 590
Completeness	p. 596
Additive Categories	p. 600
Adjoint Functors	p. 604
The Adjoint Functor Theorem	p. 609
Triples	p. 613
Tripleability	p. 616
Varieties	p. 621
Appendix	p. 625
Chain Conditions	p. 625
The Axiom of Choice	p. 628
Ordinal Numbers	p. 631
Ordinal Induction	p. 635
Cardinal Numbers	p. 639
References	p. 645
Further Readings	p. 650
Index	p. 652
Table of Contents provided by Ingram. All Rights Reserved.

Abstract Algebra

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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