Class field theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer
2009
|
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | X, 226 S. |
| ISBN: | 9780387724898 0387724893 9780387724904 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804138032309207040 |
|---|---|
| adam_text | Contents
1
A
Brief Review................................................. 1
1
Number Fields
............................................... 1
2
Completions of Number Fields
.................................. 8
3
Some General Questions Motivating Class Field Theory
............ 14
2
Dirichlet s Theorem on Primes in Arithmetic Progressions
........... 17
1 Characters of Finite Abelian Groups
............................. 17
2
Dirichlet Characters
........................................... 20
3
Dirichlet Series
............................................... 30
4
Dirichlet s Theorem on Primes in Arithmetic Progressions
.......... 35
5
Dirichlet Density
............................................. 40
3
Ray Class Groups
............................................... 45
1
The Approximation Theorem and Infinite Primes
.................. 45
2
Ray Class Groups and the Universal Norm Index Inequality
......... 47
3
The Main Theorems of Class Field Theory
........................ 60
4
The
Idèlic
Theory
............................................... 63
1
Places of a Number Field
...................................... 64
2
A Little Topology
............................................. 66
3
The Group of
Ideies
of a Number Field
........................... 68
4
Cohomology of Finite Cyclic Groups and the
Herbrand
Quotient
..... 75
5
Cyclic Galois Action on
Ideies
.................................. 83
Artin
Reciprocity
...............................................105
1
The Conductor of an Abelian Extension of Number Fields and the
Artin
Symbol
................................................ 105
2 Artin
Reciprocity
.............................................
Ill
3
An Example: Quadratic Reciprocity
.............................128
4
Some
Preliminar}
Results about the
Artin Map
on Local Fields
......130
χ
Contents
6
The Existence Theorem, Consequences and Applications
............135
1
The Ordering Theorem and the Reduction Lemma
.................136
2 Kummer
л
-extensions and the Proof of the Existence Theorem
.......139
3
The
Artin
Map on Local Fields
.................................148
4
The Hubert Class Field
........................................153
5
Arbitrary Finite Extensions of Number Fields
.....................159
6
Infinite Extensions and an Alternate Proof of the Existence Theorem
..162
7
An Example: Cyclotomic Fields
.................................168
7
Local Class Field Theory
........................................181
1
Some Preliminary Facts About Local Fields
.......................1 82
2
A Fundamental Exact Sequence
.................................186
3
Local Units Modulo Norms
....................................191
4
One-Dimensional Formal Group Laws
...........................195
5
The Formal Group Laws of
Lubin
and
Tate
.......................198
6
Lubin-Tate Extensions
........................................201
7
The Local
Artin Map..........................................210
Bibliography
.......................................................219
Index
.............................................................223
Universitext
Class Field Theory
Class field theory, the study of abelian extensions of algebraic number fields, is one of the
largest branches of algebraic number theory. It brings together the quadratic and higher
reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. Some of its
consequences (e.g., the Chebotarev density theorem) apply even to nonabelian exten¬
sions.
This book is an accessible introduction to class field theory. It takes a traditional
approach in that it presents the global material first, using some of the original tech¬
niques of proof, but in a fashion that is cleaner and more streamlined than most other
books on this topic.
It could be used for a graduate course on algebraic number theory, as well as for students
who are interested in self-study. The book has been class-tested, and the author has
included exercises throughout the text.
Professor Nancy Childress is a member of the Mathematics Faculty at Arizona State
University.
|
| adam_txt |
Contents
1
A
Brief Review. 1
1
Number Fields
. 1
2
Completions of Number Fields
. 8
3
Some General Questions Motivating Class Field Theory
. 14
2
Dirichlet's Theorem on Primes in Arithmetic Progressions
. 17
1 Characters of Finite Abelian Groups
. 17
2
Dirichlet Characters
. 20
3
Dirichlet Series
. 30
4
Dirichlet's Theorem on Primes in Arithmetic Progressions
. 35
5
Dirichlet Density
. 40
3
Ray Class Groups
. 45
1
The Approximation Theorem and Infinite Primes
. 45
2
Ray Class Groups and the Universal Norm Index Inequality
. 47
3
The Main Theorems of Class Field Theory
. 60
4
The
Idèlic
Theory
. 63
1
Places of a Number Field
. 64
2
A Little Topology
. 66
3
The Group of
Ideies
of a Number Field
. 68
4
Cohomology of Finite Cyclic Groups and the
Herbrand
Quotient
. 75
5
Cyclic Galois Action on
Ideies
. 83
Artin
Reciprocity
.105
1
The Conductor of an Abelian Extension of Number Fields and the
Artin
Symbol
. 105
2 Artin
Reciprocity
.
Ill
3
An Example: Quadratic Reciprocity
.128
4
Some
Preliminar}
Results about the
Artin Map
on Local Fields
.130
χ
Contents
6
The Existence Theorem, Consequences and Applications
.135
1
The Ordering Theorem and the Reduction Lemma
.136
2 Kummer
л
-extensions and the Proof of the Existence Theorem
.139
3
The
Artin
Map on Local Fields
.148
4
The Hubert Class Field
.153
5
Arbitrary Finite Extensions of Number Fields
.159
6
Infinite Extensions and an Alternate Proof of the Existence Theorem
.162
7
An Example: Cyclotomic Fields
.168
7
Local Class Field Theory
.181
1
Some Preliminary Facts About Local Fields
.1 82
2
A Fundamental Exact Sequence
.186
3
Local Units Modulo Norms
.191
4
One-Dimensional Formal Group Laws
.195
5
The Formal Group Laws of
Lubin
and
Tate
.198
6
Lubin-Tate Extensions
.201
7
The Local
Artin Map.210
Bibliography
.219
Index
.223
Universitext
Class Field Theory
Class field theory, the study of abelian extensions of algebraic number fields, is one of the
largest branches of algebraic number theory. It brings together the quadratic and higher
reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. Some of its
consequences (e.g., the Chebotarev density theorem) apply even to nonabelian exten¬
sions.
This book is an accessible introduction to class field theory. It takes a traditional
approach in that it presents the global material first, using some of the original tech¬
niques of proof, but in a fashion that is cleaner and more streamlined than most other
books on this topic.
It could be used for a graduate course on algebraic number theory, as well as for students
who are interested in self-study. The book has been class-tested, and the author has
included exercises throughout the text.
Professor Nancy Childress is a member of the Mathematics Faculty at Arizona State
University. |
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| index_date | 2024-07-02T22:06:54Z |
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| institution | BVB |
| isbn | 9780387724898 0387724893 9780387724904 |
| language | English |
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| spelling | Childress, Nancy Verfasser aut Class field theory Nancy Childress New York, NY Springer 2009 X, 226 S. txt rdacontent n rdamedia nc rdacarrier Universitext Class field theory Algebraischer Zahlkörper (DE-588)4068537-8 gnd rswk-swf Quadratisches Reziprozitätsgesetz (DE-588)4176566-7 gnd rswk-swf Algebraischer Zahlkörper (DE-588)4068537-8 s Quadratisches Reziprozitätsgesetz (DE-588)4176566-7 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016748643&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016748643&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Childress, Nancy Class field theory Class field theory Algebraischer Zahlkörper (DE-588)4068537-8 gnd Quadratisches Reziprozitätsgesetz (DE-588)4176566-7 gnd |
| subject_GND | (DE-588)4068537-8 (DE-588)4176566-7 |
| title | Class field theory |
| title_auth | Class field theory |
| title_exact_search | Class field theory |
| title_exact_search_txtP | Class field theory |
| title_full | Class field theory Nancy Childress |
| title_fullStr | Class field theory Nancy Childress |
| title_full_unstemmed | Class field theory Nancy Childress |
| title_short | Class field theory |
| title_sort | class field theory |
| topic | Class field theory Algebraischer Zahlkörper (DE-588)4068537-8 gnd Quadratisches Reziprozitätsgesetz (DE-588)4176566-7 gnd |
| topic_facet | Class field theory Algebraischer Zahlkörper Quadratisches Reziprozitätsgesetz |
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