Abstract algebra / Pierre Antoine Grillet.

Format:

Book

Author/Creator:

Grillet, Pierre A. (Pierre Antoine), 1941-

Series:

Graduate texts in mathematics ; v. 242.

Graduate texts in mathematics ; v. 242

Language:

English

Subjects (All):

Algebra, Abstract--Textbooks.

Algebra, Abstract.

Genre:

Textbooks.

Physical Description:

xii, 669 pages : illustrations ; 25 cm.

Edition:

Second edition.

Place of Publication:

New York : Springer, [2007]

Summary:

Abstract Algebra is a clearly written, self-contained basic algebra text for graduate students, with a generous amount of additional material that suggests the scope of contemporary algebra. The first chapters blend standard contents with a careful introduction to proofs with arrows. The last chapters, on universal algebra, and categories, including tripleability, give valuable general views of algebra. There are over 1400 exercises, at varying degrees of difficulty. For the new edition, the author has completely rewritten the entire text, streamlining the first chapters for rapid access to Galois theory, removing some material, and adding introductions to Groebner bases, Ext and Tor, and other topics.

Contents:

I Groups 1

1 Semigroups 1

2 Groups 8

3 Subgroups 12

4 Homomorphisms 18

5 The Isomorphism Theorems 23

6 Free Groups 27

7 Presentations 31

8 Free Products 37

II Structure of Groups 43

1 Direct Products 43

2 The Krull-Schmidt Theorem 48

3 Group Actions 54

4 Symmetric Groups 58

5 The Sylow Theorems 64

6 Small Groups 67

7 Composition Series 70

8 The General Linear Group 76

9 Solvable Groups 83

10 Nilpotent Groups 89

11 Semidirect Products 92

12 Group Extensions 95

III Rings 105

1 Rings 105

2 Subrings and Ideals 109

3 Homomorphisms 112

4 Domains and Fields 116

5 Polynomials in One Variable 119

6 Polynomials in Several Variables 125

7 Formal Power Series 130

8 Principal Ideal Domains 133

9 Rational Fractions 139

10 Unique Factorization Domains 141

11 Noetherian Rings 146

12 Grobner Bases 148

IV Field Extensions 155

1 Fields 155

2 Extensions 159

3 Algebraic Extensions 164

4 The Algebraic Closure 165

5 Separable Extensions 169

6 Purely Inseparable Extensions 173

7 Resultants and Discriminants 176

8 Transcendental Extensions 181

9 Separability 184

V Galois Theory 191

1 Splitting Fields 191

2 Normal Extensions 193

3 Galois Extensions 197

4 Infinite Galois Extensions 200

5 Polynomials 204

6 Cyclotomy 211

7 Norm and Trace 215

8 Solvability by Radicals 221

9 Geometric Constructions 226

VI Fields with Orders or Valuations 231

1 Ordered Fields 231

2 Real Fields 234

3 Absolute Values 239

4 Completions 243

5 Extensions 247

6 Valuations 251

7 Extending Valuations 256

8 Hensel's Lemma 261

9 Filtrations and Completions 266

VII Commutative Rings 273

1 Primary Decomposition 273

2 Ring Extensions 277

3 Integral Extensions 280

4 Localization 285

5 Dedekind Domains 290

6 Algebraic Integers 294

7 Galois Groups 297

8 Minimal Prime Ideals 300

9 Krull Dimension 304

10 Algebraic Sets 307

11 Regular Mappings 310

VIII Modules 315

2 Homomorphisms 320

3 Direct Sums and Products 324

4 Free Modules 329

5 Vector Spaces 334

6 Modules over Principal Ideal Domains 336

7 Jordan Form of Matrices 342

8 Chain Conditions 346

9 Grobner Bases 350

IX Semisimple Rings and Modules 359

1 Simple Rings and Modules 359

2 Semisimple Modules 362

3 The Artin-Wedderburn Theorem 366

4 Primitive Rings 370

5 The Jacobson Radical 374

6 Artinian Rings 377

7 Representations of Groups 380

8 Characters 386

9 Complex Characters 389

X Projectives and Injectives 393

1 Exact Sequences 393

2 Pullbacks and Pushouts 397

3 Projective Modules 401

4 Injective Modules 403

5 The Injective Hull 408

6 Hereditary Rings 411

XI Constructions 415

1 Groups of Homomorphisms 415

2 Properties of Hom 419

3 Direct Limits 423

4 Inverse Limits 429

5 Tensor Products 434

6 Properties of Tensor Products 441

7 Dual Modules 448

8 Flat Modules 450

9 Completions 456

XII Ext and Tor 463

1 Complexes 463

2 Resolutions 471

3 Derived Functors 478

4 Ext 487

5 Tor 493

6 Universal Coefficient Theorems 497

7 Cohomology of Groups 500

8 Projective Dimension 507

9 Global Dimension 510

XIII Algebras 515

1 Algebras over a Ring 515

2 The Tensor Algebra 518

3 The Symmetric Algebra 521

4 The Exterior Algebra 523

5 Tensor Products of Algebras 527

6 Tensor Products of Fields 530

7 Simple Algebras over a Field 534

XIV Lattices 539

2 Complete Lattices 543

3 Modular Lattices 545

4 Distributive Lattices 549

5 Boolean Lattices 553

XV Universal Algebra 559

1 Universal Algebras 559

2 Word Algebras 564

3 Varieties 567

4 Subdirect Products 574

XVI Categories 581

1 Definition and Examples 581

2 Functors 586

3 Limits and Colimits 590

4 Completeness 596

5 Additive Categories 600

6 Adjoint Functors 604

7 The Adjoint Functor Theorem 609

8 Triples 613

9 Tripleability 616

10 Varieties 621

1 Chain Conditions 625

2 The Axiom of Choice 628

3 Ordinal Numbers 631

4 Ordinal Induction 635

5 Cardinal Numbers 639.

Notes:

Includes bibliographical references (pages [645]-651) and index.

ISBN:

0387715673

9780387715674

0387715681

9780387715681

OCLC:

141380104