Algebraic number theory:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1997
|
| Edition: | 1. ed., 2. printing |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | 1. printing u.d.T.: Number theory ; Bd. 2 |
| Physical Description: | 269 S. |
| ISBN: | 3540630031 |
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| 100 | 1 | |a Koch, Helmut |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Algebraic number theory |c H. Koch |
| 250 | |a 1. ed., 2. printing | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1997 | |
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| 500 | |a 1. printing u.d.T.: Number theory ; Bd. 2 | ||
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Record in the Search Index
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| adam_text | H. KOCH ALGEBRAIC NUMBER THEORY SPRINGER ALGEBRAIC NUMBER FIELDS H. KOCH
CONTENTS PREFACE 7 CHAPTER 1. BASIC NUMBER THEORY 8 § 1. ORDERS IN
ALGEBRAIC NUMBER FIELDS 9 1.1. MODULES AND ORDERS 10 1.2. MODULE CLASSES
12 1.3. THE UNIT GROUP OF AN ORDER 15 1.4. THE UNIT GROUP OF A
REAL-QUADRATIC NUMBER FIELD 17 1.5. INTEGRAL REPRESENTATIONS OF RATIONAL
NUMBERS BY COMPLETE FORMS 18 1.6. BINARY QUADRATIC FORMS AND COMPLETE
MODULES IN QUADRATIC NUMBER FIELDS 19 1.7. REPRESENTATIVES FOR MODULE
CLASSES IN QUADRATIC NUMBER FIELDS 21 §2. RINGS WITH DIVISOR THEORY 22
2.1. UNIQUE FACTORIZATION IN PRIME ELEMENTS 22 2.2. THE CONCEPT OF A
DOMAIN WITH DIVISOR THEORY 23 2.3. DIVISOR THEORY FOR THE MAXIMAL ORDER
OF AN ALGEBRAIC NUMBER FIELD 25 §3. DEDEKIND RINGS 27 3.1. DEFINITION OF
DEDEKIND RINGS 28 3.2. CONGRUENCES 29 3.3. SEMILOCALIZATION 30 3.4.
EXTENSIONS OF DEDEKIND RINGS 30 3.5. DIFFERENT AND DISCRIMINANT 33 3.6.
INESSENTIAL DISCRIMINANT DIVISORS 36 3.7. NORMAL EXTENSIONS 36 3.8.
IDEALS IN ALGEBRAIC NUMBER FIELDS 40 3.9. CYCLOTOMIC FIELDS 41 3.10.
APPLICATION TO FERMAT S LAST THEOREM I 43 2 CONTENTS §4. VALUATIONS 45
4.1. DEFINITION AND FIRST PROPERTIES OF VALUATIONS 45 4.2. COMPLETION OF
A FIELD WITH RESPECT TO A VALUATION 49 4.3. COMPLETE FIELDS WITH
DISCRETE VALUATION 49 4.4. THE MULTIPLICATIVE STRUCTURE OF A P-ADIC
NUMBER FIELD 51 4.5. EXTENSION OF VALUATIONS 53 4.6. FINITE EXTENSIONS
OF P-ADIC NUMBER FIELDS 55 4.7. KUMMER EXTENSIONS 58 4.8. ANALYTIC
FUNCTIONS IN COMPLETE NON-ARCHIMEDEAN VALUED FIELDS 59 4.9. THE
ELEMENTARY FUNCTIONS IN P-ADIC ANALYSIS 60 4.10. LUBIN-TATE EXTENSIONS
62 § 5. HARMONIC ANALYSIS ON LOCAL AND GLOBAL FIELDS 63 5.1. HARMONIC
ANALYSIS ON LOCAL FIELDS, THE ADDITIVE GROUP .... 64 5.2. HARMONIC
ANALYSIS ON LOCAL FIELDS, THE MULTIPLICATIVE GROUP 65 5.3. ADELES 66
5.4. IDELES 68 5.5. SUBGROUPS OF J(K)/K* OF FINITE INDEX AND THE RAY
CLASS GROUPS 69 § 6. HECKE L-SERIES AND THE DISTRIBUTION OF PRIME IDEALS
70 6.1. THE LOCAL ZETA FUNCTIONS 74 6.2. THE GLOBAL FUNCTIONAL EQUATION
76 6.3. HECKE CHARACTERS * 77 6.4. THE FUNCTIONAL EQUATION FOR HECKE
L-SERIES 79 6.5. GAUSSIAN SUMS 80 6.6. ASYMPTOTICAL DISTRIBUTION OF
IDEALS AND PRIME IDEALS 82 6.7. CHEBOTAREV S DENSITY THEOREM 84 6.8.
KRONECKER DENSITIES AND THE THEOREM OF BAUER 85 6.9. THE PRIME IDEAL
THEOREM WITH REMAINDER TERM 87 6.10. EXPLICIT FORMULAS 87 6.11.
DISCRIMINANT ESTIMATION 88 CHAPTER 2. CLASS FIELD THEORY 90 § 1. THE
MAIN THEOREMS OF CLASS FIELD THEORY 92 1.1. CLASS FIELD THEORY FOR
ABELIAN EXTENSIONS OF Q 92 1.2. THE HILBERT CLASS FIELD 93 1.3. LOCAL
CLASS FIELD THEORY 93 1.4. THE IDELE CLASS GROUP OF A NORMAL EXTENSION
95 1.5. GLOBAL CLASS FIELD THEORY 96 1.6. THE FUNCTORIAL BEHAVIOR OF THE
NORM SYMBOL 97 1.7. ARTIN S GENERAL RECIPROCITY LAW 98 1.8. THE POWER
RESIDUE SYMBOL 99 1.9. THE HILBERT NORM SYMBOL 101 1.10. THE RECIPROCITY
LAW FOR THE POWER RESIDUE SYMBOL 102 CONTENTS 3 1.11. THE PRINCIPAL
IDEAL THEOREM 103 1.12. LOCAL-GLOBAL RELATIONS 104 1.13. THE ZETA
FUNCTION OF AN ABELIAN EXTENSION 105 § 2. COMPLEX MULTIPLICATION 107
2.1. THE MAIN POLYNOMIAL 107 2.2. THE FIRST MAIN THEOREM 108 2.3. THE
RECIPROCITY LAW 109 2.4. THE CONSTRUCTION OF THE RAY CLASS FIELD 109
2.5. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION ILL 2.6. GENERALIZATION
112 § 3. COHOMOLOGY OF GROUPS 112 3.1. DEFINITION OF COHOMOLOGY GROUPS
112 3.2. FUNCTORIALITY AND THE LONG EXACT SEQUENCE 113 3.3. DIMENSION
SHIFTING 114 3.4. SHAPIRO S LEMMA 115 3.5. CORESTRICTION 115 3.6. THE
TRANSGRESSION AND THE HOCHSCHILD-SERRE-SEQUENCE 116 3.7. CUP PRODUCT 117
3.8. MODIFIED COHOMOLOGY FOR FINITE GROUPS 119 3.9. COHOMOLOGY FOR
CYCLIC GROUPS 120 3.10. THE THEOREM OF TATE 121 § 4. PROOF OF THE MAIN
THEOREMS OF CLASS FIELD THEORY 121 4.1. APPLICATION OF THE THEOREM OF
TATE TO CLASS FIELD THEORY ... 121 4.2. CLASS FORMATIONS 122 4.3.
COHOMOLOGY OF LOCAL FIELDS 124 4.4. COHOMOLOGY OF IDELES AND IDELE
CLASSES 125 4.5. ANALYTICAL PROOF OF THE SECOND INEQUALITY 129 4.6. THE
CANONICAL CLASS FOR GLOBAL EXTENSIONS 130 § 5. SIMPLE ALGEBRAS 131 5.1.
SIMPLE ALGEBRAS OVER ARBITRARY FIELDS 131 5.2. THE REDUCED TRACE AND
NORM 132 5.3. SPLITTING FIELDS 133 5.4. THE BRAUER GROUP 133 5.5. SIMPLE
ALGEBRAS OVER LOCAL FIELDS 134 5.6. THE STRUCTURE OF THE BRAUER GROUP OF
AN ALGEBRAIC NUMBER FIELD 135 5.7. SIMPLE ALGEBRAS OVER ALGEBRAIC NUMBER
FIELDS 136 § 6. EXPLICIT RECIPROCITY LAWS AND SYMBOLS 137 6.1. THE
EXPLICIT RECIPROCITY LAW OF SHAFAREVICH 138 6.2. THE EXPLICIT
RECIPROCITY LAW OF BRUCKNER AND VOSTOKOV 139 6.3. APPLICATION TO
FERMAT S LAST THEOREM II 141 6.4. SYMBOLS 142 6.5. SYMBOLS OF P-ADIC
NUMBER FIELDS 143 6.6. TAME AND WILD SYMBOLS 144 6.7. REMARKS ABOUT
MILNOR S K-THEORY 144 4 CONTENTS § 7. FURTHER RESULTS OF CLASS FIELD
THEORY 145 7.1. THE THEOREM OF SHAFAREVICH-WEIL 145 7.2. UNIVERSAL NORMS
145 7.3. ON THE STRUCTURE OF THE IDEAL CLASS GROUP 146 7.4. LEOPOLDT S
SPIEGELUNGSSATZ 147 7.5. THE COHOMOLOGY OF THE MULTIPLICATIVE GROUP 149
CHAPTER 3. GALOIS GROUPS 150 § 1. COHOMOLOGY OF PROFINITE GROUPS 151
1.1. INVERSE LIMITS OF GROUPS AND RINGS 151 1.2. PROFINITE GROUPS 153
1.3. SUPERNATURAL NUMBERS 154 1.4. PRO-P-GROUPS AND P-SYLOW GROUPS 154
1.5. FREE PROFINITE, FREE PROSOLVABLE, AND FREE PRO-P-GROUPS .... 154
1.6. DISCRETE MODULES 155 1.7. INDUCTIVE LIMITS IN C 156 1.8. GALOIS
THEORY OF INFINITE ALGEBRAIC EXTENSIONS 157 1.9. COHOMOLOGY OF PROFINITE
GROUPS 159 1.10. COHOMOLOGICAL DIMENSION 159 1.11. THE DUALIZING MODULE
160 1.12. COHOMOLOGY OF PRO-P-GROUPS 161 1.13. PRESENTATION OF
PRO-P-GROUPS BY MEANS OF GENERATORS AND RELATIONS 162 1.14. POINCARE
GROUPS 164 1.15. THE STRUCTURE OF THE RELATIONS AND THE CUP PRODUCT 165
1.16. GROUP RINGS AND THE THEOREM OF GOLOD-SHAFAREVICH 166 §2. GALOIS
COHOMOLOGY OF LOCAL AND GLOBAL FIELDS 168 2.1. EXAMPLES OF GALOIS
COHOMOLOGY OF ARBITRARY FIELDS 168 2.2. THE ALGEBRAIC CLOSURE OF A LOCAL
FIELD 169 2.3. THE MAXIMAL P-EXTENSION OF A LOCAL FIELD 171 2.4. THE
GALOIS GROUP OF A LOCAL FIELD 173 2.5. THE MAXIMAL ALGEBRAIC EXTENSION
WITH GIVEN RAMIFICATION .. 175 2.6. THE MAXIMAL P-EXTENSION WITH GIVEN
RAMIFICATION 177 2.7. THE CLASS FIELD TOWER PROBLEM 180 2.8.
DISCRIMINANT ESTIMATION FROM ABOVE 181 2.9. CHARACTERIZATION OF AN
ALGEBRAIC NUMBER FIELD BY ITS GALOIS GROUP 182 § 3. EXTENSIONS WITH
GIVEN GALOIS GROUPS 182 3.1. EMBEDDING PROBLEMS 183 3.2. EMBEDDING
PROBLEMS FOR LOCAL AND GLOBAL FIELDS 185 3.3. EXTENSIONS WITH PRESCRIBED
GALOIS GROUP OF /-POWER ORDER .. 186 3.4. EXTENSIONS WITH PRESCRIBED
SOLVABLE GALOIS GROUP 188 3.5. EXTENSIONS WITH PRESCRIBED LOCAL BEHAVIOR
188 3.6. REALIZATION OF EXTENSIONS WITH PRESCRIBED GALOIS GROUP BY MEANS
OF HILBERT S IRREDUCIBILITY THEOREM 190 CONTENTS 5 CHAPTER 4. ABELIAN
FIELDS 192 § 1. THE INTEGERS OF AN ABELIAN FIELD 193 1.1. THE
COORDINATES 193 1.2. THE GALOIS MODULE STRUCTURE OF THE RING OF INTEGERS
OF AN ABELIAN FIELD 194 §2. THE ARITHMETICAL CLASS NUMBER FORMULA 195
2.1. THE ARITHMETICAL CLASS NUMBER FORMULA FOR COMPLEX ABELIAN FIELDS
195 2.2. THE ARITHMETICAL CLASS NUMBER FORMULA FOR REAL QUADRATIC FIELDS
197 2.3. THE ARITHMETICAL CLASS NUMBER FORMULA FOR REAL ABELIAN FIELDS
198 2.4. THE STICKELBERGER IDEAL 200 2.5. ON THE P-COMPONENT OF THE
CLASS GROUP OF Q(C PM ) 203 2.6. APPLICATION TO FERMAT S LAST THEOREM
III 205 § 3. IWASAWA S THEORY OF T-EXTENSIONS 206 3.1. CLASS FIELD
THEORY OF / -EXTENSIONS 206 3.2. THE STRUCTURE OF /1-MODULES 207 3.3.
THE P-CLASS GROUP OF A ./ -EXTENSION 208 3.4. IWASAWA S THEOREM 209 §4.
P-ADIC L-FUNCTIONS 211 4.1. THE HURWITZ ZETA FUNCTION 212 4.2. P-ADIC
L-FUNCTIONS 213 4.3. CONGRUENCES FOR BERNOULLI NUMBERS 214 4.4.
GENERALIZATION TO TOTALLY REAL NUMBER FIELDS 215 4.5. THE P-ADIC CLASS
NUMBER FORMULA 215 4.6. IWASAWA S CONSTRUCTION OF P-ADIC L-FUNCTIONS 216
4.7. THE MAIN CONJECTURE 218 CHAPTER 5. ARTIN L-FUNCTIONS AND GALOIS
MODULE STRUCTURE 219 § 1. ARTIN L-FUNCTIONS 222 1.1. REPRESENTATIONS OF
FINITE GROUPS 222 1.2. ARTIN L-FUNCTIONS 224 1.3. CYCLOTOMIC FIELDS WITH
CLASS NUMBER 1 226 1.4. IMAGINARY-QUADRATIC FIELDS WITH SMALL CLASS
NUMBER 227 1.5. THE ARTIN REPRESENTATION AND THE ARTIN CONDUCTOR 228
1.6. THE FUNCTIONAL EQUATION FOR ARTIN L-FUNCTIONS 230 1.7. THE
CONJECTURES OF STARK ABOUT ARTIN L-FUNCTIONS AT S = 0 .. 231 § 2. GALOIS
MODULE STRUCTURE AND ARTIN ROOT NUMBERS 234 2.1. THE CLASS GROUP OF Z[G]
235 2.2. THE GALOIS MODULE STRUCTURE OF TAME EXTENSIONS 236 2.3. FURTHER
RESULTS ON GALOIS MODULE STRUCTURE 236 6 CONTENTS APPENDIX 1. FIELDS,
DOMAINS, AND COMPLEXES 237 1.1. FINITE FIELD EXTENSIONS 237 1.2. GALOIS
THEORY 238 1.3. DOMAINS 238 1.4. COMPLEXES 239 APPENDIX 2. QUADRATIC
RESIDUES 240 APPENDIX 3. LOCALLY COMPACT GROUPS 241 3.1. LOCALLY COMPACT
ABELIAN GROUPS 241 3.2. RESTRICTED PRODUCTS 243 APPENDIX 4. BERNOULLI
NUMBERS 243 TABLES 245 REFERENCES 251 AUTHOR INDEX 263 SUBJECT INDEX 266
|
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| id | DE-604.BV011407791 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:09:15Z |
| institution | BVB |
| isbn | 3540630031 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007669897 |
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| owner_facet | DE-703 DE-29 DE-29T DE-824 DE-355 DE-BY-UBR DE-521 DE-19 DE-BY-UBM DE-83 DE-11 |
| physical | 269 S. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
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| publisher | Springer |
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| spelling | Koch, Helmut Verfasser aut Algebraic number theory H. Koch 1. ed., 2. printing Berlin [u.a.] Springer 1997 269 S. txt rdacontent n rdamedia nc rdacarrier 1. printing u.d.T.: Number theory ; Bd. 2 Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007669897&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Koch, Helmut Algebraic number theory Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| subject_GND | (DE-588)4001170-7 |
| title | Algebraic number theory |
| title_auth | Algebraic number theory |
| title_exact_search | Algebraic number theory |
| title_full | Algebraic number theory H. Koch |
| title_fullStr | Algebraic number theory H. Koch |
| title_full_unstemmed | Algebraic number theory H. Koch |
| title_short | Algebraic number theory |
| title_sort | algebraic number theory |
| topic | Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| topic_facet | Algebraische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007669897&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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