Algebraic number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1994
|
| Edition: | 2. ed. |
| Series: | Graduate texts in mathematics
110 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 357 S. graph. Darst. |
| ISBN: | 3540942254 0387942254 |
Staff View
MARC
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| 100 | 1 | |a Lang, Serge |d 1927-2005 |e Verfasser |0 (DE-588)119305119 |4 aut | |
| 245 | 1 | 0 | |a Algebraic number theory |c Serge Lang |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York u.a. |b Springer |c 1994 | |
| 300 | |a XIII, 357 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 110 | |
| 650 | 4 | |a Nombres algébriques, Théorie des | |
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| 650 | 4 | |a Algebraic number theory | |
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Record in the Search Index
| _version_ | 1804123897321226240 |
|---|---|
| adam_text | Contents
Part One
General Basic Theory
Chapter I
Algebraic Integers
1. Localization 3
2. Integral closure 4
3. Prime ideals 8
4. Chinese remainder theorem 11
5. Galois extensions 12
6. Dedekind rings 18
7. Discrete valuation rings 22
8. Explicit factorization of a prime 27
9. Projective modules over Dedekind rings 29
Chapter II
Completions
1. Definitions and completions 31
2. Polynomials in complete fields 41
3. Some filtrations 45
4. Unramified extensions 48
5. Tamely ramified extensions 51
Chapter III
The Different and Discriminant
1. Complementary modules 57
2. The different and ramification 62
3. The discriminant 64
ix
X CONTENTS
Chapter IV
Cyclotomic Fields
1. Roots of unity 71
2. Quadratic fields 76
3. Gauss sums 82
4. Relations in ideal classes 96
Chapter V
Parallelotopes
1. The product formula 99
2. Lattice points in parallelotopes 110
3. A volume computation 116
4. Minkowski s constant 119
Chapter VI
The Ideal Function
1. Generalized ideal classes 123
2. Lattice points in homogeneously expanding domains 128
3. The number of ideals in a given class 129
Chapter VII
Ideles and Adeles
1. Restricted direct products 137
2. Adeles 139
3. Ideles 140
4. Generalized ideal class groups; relations with idele classes . . . .145
5. Embedding of k* in the idele classes 151
6. Galois operation on ideles and idele classes 152
Chapter VIII
Elementary Properties of the Zeta Function and L series
1. Lemmas on Dirichlet series 155
2. Zeta function of a number field 159
3. The L series 162
4. Density of primes in arithmetic progressions 166
5. Faltings finiteness theorem 170
CONTENTS xi
Part Two
Class Field Theory
Chapter IX
Norm Index Computations
1. Algebraic preliminaries 179
2. Exponential and logarithm functions 185
3. The local norm index 187
4. A theorem on units 190
5. The global cyclic norm index 193
6. Applications 195
Chapter X
The Artin Symbol, Reciprocity Law, and Class Field Theory
1. Formalism of the Artin symbol 197
2. Existence of a conductor for the Artin symbol 200
3. Class fields 206
Chapter XI
The Existence Theorem and Local Class Field Theory
1. Reduction to Kummer extensions 213
2. Proof of the existence theorem 215
3. The complete splitting theorem 217
4. Local class field theory and the ramification theorem 219
5. The Hilbert class field and the principal ideal theorem 224
6. Infinite divisibility of the universal norms 226
Chapter XII
L series Again
1. The proper abelian L series 229
2. Artin (non abelian) L series 232
3. Induced characters and L series contributions 236
xii CONTENTS
Part Three
Analytic Theory
Chapter XIII
Functional Equation of the Zeta Function, Hecke s Proof
1. The Poisson summation formula 245
2. A special computation 250
3. Functional equation 253
4. Application to the Brauer Siegel theorem 260
5. Applications to the ideal function 262
Appendix: Other applications 273
Chapter XIV
Functional Equation, Tate s Thesis
1. Local additive duality 276
2. Local multiplicative theory 278
3. Local functional equation 280
4. Local computations 282
5. Restricted direct products 287
6. Global additive duality and Riemann Roch theorem 289
7. Global functional equation 292
8. Global computations 297
Chapter XV
Density of Primes and Tauberian Theorem
1. The Dirichlet integral 303
2. Ikehara s Tauberian theorem 304
3. Tauberian theorem for Dirichlet series 310
4. Non vanishing of the L series 312
5. Densities 315
Chapter XVI
The Brauer Siegel Theorem
1. An upper estimate for the residue 322
2. A lower bound for the residue 323
3. Comparison of residues in normal extensions 325
4. End of the proofs 327
Appendix: Brauer s lemma 328
CONTENTS xiii
Chapter XVII
Explicit Formulas
1. Weierstrass factorization of the L series 331
2. An estimate for £ /£ 333
3. The Weil formula 337
4. The basic sum and the first part of its evaluation 344
5. Evaluation of the sum: Second part 348
Bibliography 353
Index 355
|
| any_adam_object | 1 |
| author | Lang, Serge 1927-2005 |
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| classification_rvk | SK 180 |
| classification_tum | MAT 120f |
| ctrlnum | (OCoLC)29667945 (DE-599)BVBBV009555986 |
| dewey-full | 512/.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.74 |
| dewey-search | 512/.74 |
| dewey-sort | 3512 274 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV009555986 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:37:04Z |
| institution | BVB |
| isbn | 3540942254 0387942254 |
| language | English |
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| physical | XIII, 357 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Algebraic number theory Serge Lang 2. ed. New York u.a. Springer 1994 XIII, 357 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 110 Nombres algébriques, Théorie des Nombres algébriques, théorie des ram Algebraic number theory Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 s DE-604 Analytische Zahlentheorie (DE-588)4001870-2 s 1\p DE-604 Graduate texts in mathematics 110 (DE-604)BV000000067 110 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006314184&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lang, Serge 1927-2005 Algebraic number theory Graduate texts in mathematics Nombres algébriques, Théorie des Nombres algébriques, théorie des ram Algebraic number theory Algebraische Zahlentheorie (DE-588)4001170-7 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4001170-7 (DE-588)4001870-2 |
| title | Algebraic number theory |
| title_auth | Algebraic number theory |
| title_exact_search | Algebraic number theory |
| title_full | Algebraic number theory Serge Lang |
| title_fullStr | Algebraic number theory Serge Lang |
| title_full_unstemmed | Algebraic number theory Serge Lang |
| title_short | Algebraic number theory |
| title_sort | algebraic number theory |
| topic | Nombres algébriques, Théorie des Nombres algébriques, théorie des ram Algebraic number theory Algebraische Zahlentheorie (DE-588)4001170-7 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Nombres algébriques, Théorie des Nombres algébriques, théorie des Algebraic number theory Algebraische Zahlentheorie Analytische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006314184&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT langserge algebraicnumbertheory |