The theory of classical valuations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1999
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 403 S. |
| ISBN: | 0387985255 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface v
Notations xi
CHAPTER 0
Introduction 1
CHAPTER 1
Absolute Values of Fields 3
1.1. First Examples 3
1.2. Generalities About Absolute Values of a Field 6
1.3. Absolute Values of Q 13
1.4. The Independence of Absolute Values 16
1.5. The Topology of Valued Fields 19
1.6. Archimedean Absolute Values 32
1.7. Topological Characterizations of Valued Fields 43
CHAPTER 2
Valuations of a Field 55
2.1. Generalities About Valuations of a Field 55
2.2. Complete Valued Fields and S)p 67
CHAPTER 3
Polynomials and Henselian Valued Fields 79
3.1. Polynomials over Valued Fields 79
3.2. Henselian Valued Fields 94
CHAPTER 4
Extensions of Valuations 107
4.1. Existence of Extensions and General Results 107
4.2. The Set of Extensions of a Valuation 118
viii Contents
CHAPTER 5
Uniqueness of Extensions of Valuations and
Poly Complete Fields 127
5.1. Uniqueness of Extensions 127
5.2. Poly Complete Fields 146
CHAPTER 6
Extensions of Valuations: Numerical Relations 151
6.1. Numerical Relations for Valuations with Unique Extension 151
6.2. Numerical Relations in the General Case 158
6.3. Some Interesting Examples 172
6.4. Appendix on/7 Groups 178
CHAPTER 7
Power Series and the Structure of Complete Valued Fields 185
7.1. Power Series 185
7.2. Structure of Complete Discrete Valued Fields 193
CHAPTER 8
Decomposition and Inertia Theory 213
8.1. Decomposition Theory 213
8.2. Inertia Theory 217
CHAPTER 9
Ramification Theory 227
9.1. Lower Ramification Theory 227
9.2. Higher Ramification 244
CHAPTER 10
Valuation Characterizations of Dedekind Domains 263
10.1. Valuation Properties of the Rings of Algebraic Integers 263
10.2. Characterizations of Dedekind Domains 280
10.3. Characterizations of Valuation Domains 288
CHAPTER 11
Galois Groups of Algebraic Extensions of Infinite Degree 307
11.1. Galois Extensions of Infinite Degree 307
11.2. The Abelian Closure of Q 320
Contents ix
CHAPTER 12
Ideals, Valuations, and Divisors in Algebraic
Extensions of Infinite Degree over (Q 333
12.1. Ideals 334
12.2. Valuations, Dedekind Domains, and Examples 339
12.3. Divisors of Algebraic Number Fields of Infinite Degree 356
CHAPTER 13
A Glimpse of Krull Valuations 365
13.1. Generalities 365
13.2. Integrally Closed Domains 369
13.3. Suggestions for Further Study 372
APPENDIX
Commutative Fields and Characters of
Finite Abelian Groups 373
A.I. Algebraic Elements 373
A. 2. Algebraic Elements, Algebraically Closed Fields 374
A.3. Algebraic Number Fields 375
A.4. Characteristic and Prime Fields 375
A.5. Normal Extensions and Splitting Fields 376
A.6. Separable Extensions 377
A.7. Galois Extensions 378
A.8. Roots of Unity 379
A.9. Finite Fields 381
A. 10. Trace and Norm of Elements 381
A. 11. The Discriminant 382
A. 12. Discriminant and Resultant of Polynomials 383
A. 13. Inseparable Extensions 385
A. 14. Perfect Fields 386
A. 15. The Theorem of Steinitz 386
A. 16. Orderable Fields 387
A. 17. The Theorem of Artin 388
A. 18. Characters of Finite Abelian Groups 388
Bibliography 391
Index 395
|
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| isbn | 0387985255 |
| language | English |
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| spelling | Ribenboim, Paulo 1928- Verfasser (DE-588)117719404 aut The theory of classical valuations Paulo Ribenboim New York [u.a.] Springer 1999 IX, 403 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Corps algébriques Valuations, Théorie des Algebraic fields Valuation theory Bewertungstheorie (DE-588)4122918-6 gnd rswk-swf Algebraischer Körper (DE-588)4141852-9 gnd rswk-swf Bewertungstheorie (DE-588)4122918-6 s DE-604 Algebraischer Körper (DE-588)4141852-9 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008378055&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ribenboim, Paulo 1928- The theory of classical valuations Corps algébriques Valuations, Théorie des Algebraic fields Valuation theory Bewertungstheorie (DE-588)4122918-6 gnd Algebraischer Körper (DE-588)4141852-9 gnd |
| subject_GND | (DE-588)4122918-6 (DE-588)4141852-9 |
| title | The theory of classical valuations |
| title_auth | The theory of classical valuations |
| title_exact_search | The theory of classical valuations |
| title_full | The theory of classical valuations Paulo Ribenboim |
| title_fullStr | The theory of classical valuations Paulo Ribenboim |
| title_full_unstemmed | The theory of classical valuations Paulo Ribenboim |
| title_short | The theory of classical valuations |
| title_sort | the theory of classical valuations |
| topic | Corps algébriques Valuations, Théorie des Algebraic fields Valuation theory Bewertungstheorie (DE-588)4122918-6 gnd Algebraischer Körper (DE-588)4141852-9 gnd |
| topic_facet | Corps algébriques Valuations, Théorie des Algebraic fields Valuation theory Bewertungstheorie Algebraischer Körper |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008378055&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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