Abstract algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer
2007
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
242 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | 1. Aufl. u.d.T.: Grillet, Pierre A.: Algebra. - Literaturverz.: S. 645 - 649 |
| Beschreibung: | XII, 669 S. graph. Darst. |
| ISBN: | 0387715673 9780387715674 9781441924506 |
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| 100 | 1 | |a Grillet, Pierre A. |d 1941- |e Verfasser |0 (DE-588)133555186 |4 aut | |
| 245 | 1 | 0 | |a Abstract algebra |c Pierre Antoine Grillet |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY |b Springer |c 2007 | |
| 300 | |a XII, 669 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 242 | |
| 500 | |a 1. Aufl. u.d.T.: Grillet, Pierre A.: Algebra. - Literaturverz.: S. 645 - 649 | ||
| 650 | 0 | 7 | |a Universelle Algebra |0 (DE-588)4061777-4 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804137152689209344 |
|---|---|
| adam_text | Contents
Preface vii
Starred sections and chapters may be skipped at first reading.
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
x Contents
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
Contents xi
11. Regular Mappings 310
VIII. Modules 315
1. Definition 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 Horn 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
xii Contents
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
1. Definitions 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
A. Appendix 625
1. Chain Conditions 625
2. The Axiom of Choice 628
3. Ordinal Numbers 631
4. Ordinal Induction 635
5. Cardinal Numbers 639
References 645
Further Readings 650
Index 652
|
| adam_txt |
Contents
Preface vii
Starred sections and chapters may be skipped at first reading.
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
x Contents
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
Contents xi
11. Regular Mappings 310
VIII. Modules 315
1. Definition 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 Horn 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
xii Contents
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
1. Definitions 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
A. Appendix 625
1. Chain Conditions 625
2. The Axiom of Choice 628
3. Ordinal Numbers 631
4. Ordinal Induction 635
5. Cardinal Numbers 639
References 645
Further Readings 650
Index 652 |
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| author | Grillet, Pierre A. 1941- |
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| classification_tum | MAT 110f MAT 080f |
| ctrlnum | (OCoLC)255964387 (DE-599)BVBBV022886496 |
| dewey-full | 512.02 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.02 |
| dewey-search | 512.02 |
| dewey-sort | 3512.02 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| index_date | 2024-07-02T18:52:07Z |
| indexdate | 2024-07-09T21:07:45Z |
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| isbn | 0387715673 9780387715674 9781441924506 |
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| spelling | Grillet, Pierre A. 1941- Verfasser (DE-588)133555186 aut Abstract algebra Pierre Antoine Grillet 2. ed. New York, NY Springer 2007 XII, 669 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 242 1. Aufl. u.d.T.: Grillet, Pierre A.: Algebra. - Literaturverz.: S. 645 - 649 Universelle Algebra (DE-588)4061777-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Universelle Algebra (DE-588)4061777-4 s DE-604 Erscheint auch als Online-Ausgabe 978-0-387-71568-1 Graduate texts in mathematics 242 (DE-604)BV000000067 242 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091404&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4061777-4 (DE-588)4123623-3 |
| title | Abstract algebra |
| title_auth | Abstract algebra |
| title_exact_search | Abstract algebra |
| title_exact_search_txtP | Abstract algebra |
| title_full | Abstract algebra Pierre Antoine Grillet |
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