Algebraic number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
[2000]
|
| Edition: | 2. ed., corr. 3. printing |
| Series: | Graduate texts in mathematics
110 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 353 - 354. - Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XIII, 357 S. |
| ISBN: | 0387942254 9780387942254 9781461269229 3540942254 |
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MARC
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| 490 | 1 | |a Graduate texts in mathematics |v 110 | |
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Record in the Search Index
| _version_ | 1804136145171251200 |
|---|---|
| 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 nitrations
................ 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
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 kf in the idele classes
.......... 151
6.
Galois operation on
ideies
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 »
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
¿-series Again
1.
The proper abelian L-series
............229
2. Artin (non-abelian)
L-series
............232
3.
Induced characters and L-series contributions
.......236
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
Xl»
Chapter
XVII
Explicit Formulas
1.
Weierstrass
factorization of the L-series
......... 331
2.
An estimate for
ξ /ξ
............... 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
|
| adam_txt |
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 nitrations
. 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
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 kf in the idele classes
. 151
6.
Galois operation on
ideies
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 »
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
¿-series Again
1.
The proper abelian L-series
.229
2. Artin (non-abelian)
L-series
.232
3.
Induced characters and L-series contributions
.236
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
Xl»
Chapter
XVII
Explicit Formulas
1.
Weierstrass
factorization of the L-series
. 331
2.
An estimate for
ξ'/ξ
. 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 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| id | DE-604.BV022170518 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:19:51Z |
| indexdate | 2024-07-09T20:51:44Z |
| institution | BVB |
| isbn | 0387942254 9780387942254 9781461269229 3540942254 |
| language | English |
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| physical | XIII, 357 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| 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., corr. 3. printing New York, NY [u.a.] Springer [2000] XIII, 357 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 110 Literaturverz. S. 353 - 354. - Hier auch später erschienene, unveränderte Nachdrucke Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 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 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015385221&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 Analytische Zahlentheorie (DE-588)4001870-2 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| subject_GND | (DE-588)4001870-2 (DE-588)4001170-7 |
| title | Algebraic number theory |
| title_auth | Algebraic number theory |
| title_exact_search | Algebraic number theory |
| title_exact_search_txtP | 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 | Analytische Zahlentheorie (DE-588)4001870-2 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| topic_facet | Analytische Zahlentheorie Algebraische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015385221&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT langserge algebraicnumbertheory |