Brauer type embedding problems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2005]
|
| Schriftenreihe: | Fields Institute monographs
21 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus dem Vorwort: "This monograph grew- in a slow and round-about way-out o f my Ph.D.-thesis (University o f Copenhagen, 1996) . " |
| Beschreibung: | vii, 171 Seiten |
| ISBN: | 0821837265 9780821837269 |
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Datensatz im Suchindex
| _version_ | 1807864633704316928 |
|---|---|
| adam_text |
Contents
Introduction v
Acknowledgments viii
Chapter 1. Galois Theory 1
Notation 1
1.1. Integral elements 1
1.2. Algebraic extensions and splitting fields 2
1.3. Separability 1
1.4. Galois extensions 6
1.5. The Galois group as permutation group 9
1.6. Reduction modulo primes 11
1.7. Cyclotomic fields 13
1.8. Cyclic extensions and Hilbert Satz 90 15
1.9. Radical extensions and solvability 17
1.10. The normal basis theorem 19
Exercises 21
Chapter 2. Inverse Galois Theory and Embedding Problems 27
2.1. Inverse Galois theory 27
2.2. Definition of Galois theoretical embedding problems 29
2.3. Group cohomology 30
2.4. Brauer type embedding problems 34
Exercises 40
Chapter 3. Brauer Groups 43
3.1. Basic facts about algebras 43
3.2. Simple Artinian rings 44
3.3. Finite dimensional central simple algebras 46
3.4. The Brauer group 50
3.5. Cyclic algebras 53
3.6. Brauer type embedding problems revisited 56
3.7. Extensions of the dihedral group 59
Exercises 62
Chapter 4. Group Cohomology 67
4.1. The cohomologies in dimensions 0 and 1 67
4.2. Galois Twist 68
4.3. The long exact cohomology sequence 70
4.4. The corestriction map 72
iii
iv Contents
Exercises 73
Chapter 5. Quadratic Forms 77
5.1. Basic theory of quadratic forms 77
5.2. Clifford algebras 81
5.3. Quadratic forms and non abelian cohomology 83
Exercises 85
Chapter 6. Decomposing the Obstruction 87
6.1. Decomposition along a direct product 87
6.2. Orthogonal representations 95
6.3. Projective representations 98
Exercises 102
Chapter 7. Quadratic Forms and Embedding Problems 105
7.1. Groups of exponent 4 105
7.2. Witt's Criterion 108
7.3. Groups of exponent 8 114
Exercises 121
Chapter 8. Reducing the Embedding Problem 123
8.1. Reduction to Brauer type embedding problems 123
8.2. Examples: The holomorph of the quaternion group 133
8.3. Embedding problems with cyclic kernel of order 4 141
Exercises 148
Appendix A. Pro finite Galois Theory 151
Introduction 151
A.I. The separable closure of a field 151
A.2. Galois extensions 152
A.3. Pro finite groups 154
A.4. Pro finite Galois theory 156
A.5. Pro finite group cohomology 157
A.6. Pro finite cohomology and the Brauer group 158
Exercises 160
Bibliography 163
Index 167 |
| adam_txt |
Contents
Introduction v
Acknowledgments viii
Chapter 1. Galois Theory 1
Notation 1
1.1. Integral elements 1
1.2. Algebraic extensions and splitting fields 2
1.3. Separability 1
1.4. Galois extensions 6
1.5. The Galois group as permutation group 9
1.6. Reduction modulo primes 11
1.7. Cyclotomic fields 13
1.8. Cyclic extensions and Hilbert Satz 90 15
1.9. Radical extensions and solvability 17
1.10. The normal basis theorem 19
Exercises 21
Chapter 2. Inverse Galois Theory and Embedding Problems 27
2.1. Inverse Galois theory 27
2.2. Definition of Galois theoretical embedding problems 29
2.3. Group cohomology 30
2.4. Brauer type embedding problems 34
Exercises 40
Chapter 3. Brauer Groups 43
3.1. Basic facts about algebras 43
3.2. Simple Artinian rings 44
3.3. Finite dimensional central simple algebras 46
3.4. The Brauer group 50
3.5. Cyclic algebras 53
3.6. Brauer type embedding problems revisited 56
3.7. Extensions of the dihedral group 59
Exercises 62
Chapter 4. Group Cohomology 67
4.1. The cohomologies in dimensions 0 and 1 67
4.2. Galois Twist 68
4.3. The long exact cohomology sequence 70
4.4. The corestriction map 72
iii
iv Contents
Exercises 73
Chapter 5. Quadratic Forms 77
5.1. Basic theory of quadratic forms 77
5.2. Clifford algebras 81
5.3. Quadratic forms and non abelian cohomology 83
Exercises 85
Chapter 6. Decomposing the Obstruction 87
6.1. Decomposition along a direct product 87
6.2. Orthogonal representations 95
6.3. Projective representations 98
Exercises 102
Chapter 7. Quadratic Forms and Embedding Problems 105
7.1. Groups of exponent 4 105
7.2. Witt's Criterion 108
7.3. Groups of exponent 8 114
Exercises 121
Chapter 8. Reducing the Embedding Problem 123
8.1. Reduction to Brauer type embedding problems 123
8.2. Examples: The holomorph of the quaternion group 133
8.3. Embedding problems with cyclic kernel of order 4 141
Exercises 148
Appendix A. Pro finite Galois Theory 151
Introduction 151
A.I. The separable closure of a field 151
A.2. Galois extensions 152
A.3. Pro finite groups 154
A.4. Pro finite Galois theory 156
A.5. Pro finite group cohomology 157
A.6. Pro finite cohomology and the Brauer group 158
Exercises 160
Bibliography 163
Index 167 |
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| spelling | Ledet, Arne Verfasser aut Brauer type embedding problems Arne Ledet Providence, Rhode Island American Mathematical Society [2005] © 2005 vii, 171 Seiten txt rdacontent n rdamedia nc rdacarrier Fields Institute monographs 21 Aus dem Vorwort: "This monograph grew- in a slow and round-about way-out o f my Ph.D.-thesis (University o f Copenhagen, 1996) . " Brauer, Groupes de Galois, Théorie inverse de Inverse Galois theory Brauer groups Brauer-Gruppe (DE-588)4146488-6 gnd rswk-swf Inverses Problem (DE-588)4125161-1 gnd rswk-swf Galois-Theorie (DE-588)4155901-0 gnd rswk-swf Galois-Theorie (DE-588)4155901-0 s Inverses Problem (DE-588)4125161-1 s Brauer-Gruppe (DE-588)4146488-6 s b DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-3148-8 Fields Institute monographs 21 (DE-604)BV009737926 21 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014167597&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ledet, Arne Brauer type embedding problems Fields Institute monographs Brauer, Groupes de Galois, Théorie inverse de Inverse Galois theory Brauer groups Brauer-Gruppe (DE-588)4146488-6 gnd Inverses Problem (DE-588)4125161-1 gnd Galois-Theorie (DE-588)4155901-0 gnd |
| subject_GND | (DE-588)4146488-6 (DE-588)4125161-1 (DE-588)4155901-0 |
| title | Brauer type embedding problems |
| title_auth | Brauer type embedding problems |
| title_exact_search | Brauer type embedding problems |
| title_exact_search_txtP | Brauer type embedding problems |
| title_full | Brauer type embedding problems Arne Ledet |
| title_fullStr | Brauer type embedding problems Arne Ledet |
| title_full_unstemmed | Brauer type embedding problems Arne Ledet |
| title_short | Brauer type embedding problems |
| title_sort | brauer type embedding problems |
| topic | Brauer, Groupes de Galois, Théorie inverse de Inverse Galois theory Brauer groups Brauer-Gruppe (DE-588)4146488-6 gnd Inverses Problem (DE-588)4125161-1 gnd Galois-Theorie (DE-588)4155901-0 gnd |
| topic_facet | Brauer, Groupes de Galois, Théorie inverse de Inverse Galois theory Brauer groups Brauer-Gruppe Inverses Problem Galois-Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014167597&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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