Problems in algebraic number theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2005
|
| Edition: | 2. ed. |
| Series: | Graduate texts in mathematics
190 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 352 S. |
| ISBN: | 0387221824 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804133010080006144 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
vii
Preface to the First Edition
ix
Acknowledgments
xi
I Problems
1
Elementary Number Theory
3
1.1
Integers
............................. 3
1.2
Applications of Unique Factorization
............. 8
1.3
The ABC Conjecture
..................... 9
1.4
Supplementary Problems
.................... 10
2
Euclidean Rings
13
2.1
Preliminaries
.......................... 13
2.2
Gaussian Integers
........................ 17
2.3 Eisenstein
Integers
....................... 17
2.4
Some Further Examples
.................... 21
2.5
Supplementary Problems
.................... 25
3
Algebraic Numbers and Integers
27
3.1
Basic Concepts
......................... 27
3.2
Liouville s Theorem and Generalizations
........... 29
3.3
Algebraic Number Fields
.................... 32
3.4
Supplementary Problems
.................... 38
4
Integral Bases
41
4.1
The Norm and the Trace
.................... 41
4.2
Existence of an Integral Basis
................. 43
4.3
Examples
............................ 46
4.4
Ideals in
Ок
........................... 49
4.5
Supplementary Problems
.................... 50
xiii
xiv CONTENTS
5 Dedekind Domains 53
5.1 Integral
Closure
......................... 53
5.2
Characterizing
Dedekind Domains.............. 55
5.3
Fractional
Ideals
and Unique Factorization..........
57
5.4
Dedekinďs
Theorem...................... 63
5.5
Factorization
in
Οχ
...................... 65
5.6
Supplementary Problems
.................... 66
6
The Ideal Class Group
69
6.1
Elementary Results
....................... 69
6.2
Finiteness of the Ideal Class Group
.............. 71
6.3
Diophantine Equations
..................... 73
6.4
Exponents of Ideal Class Groups
............... 75
6.5
Supplementary Problems
.................... 76
7
Quadratic Reciprocity
81
7.1
Preliminaries
.......................... 81
7.2
Gauss Sums
........................... 84
7.3
The Law of Quadratic Reciprocity
.............. 86
7.4
Quadratic Fields
......................... 88
7.5
Primes in Special Progressions
................ 91
7.6
Supplementary Problems
...............■..... 94
8
The Structure of Units
99
8.1
Dirichlet s Unit Theorem
................... 99
8.2
Units in Real Quadratic Fields
................ 108
8.3
Supplementary Problems
.................... 115
9
Higher Reciprocity Laws
117
9.1
Cubic Reciprocity
....................... 117
9.2 Eisenstein
Reciprocity
..................... 122
9.3
Supplementary Problems
.................... 125
10
Analytic Methods
127
10.1
The Riemann and Dedekind
Zeta
Functions
......... 127
10.2
Zeta
Functions of Quadratic Fields
.............. 130
10.3
Dirichlet s ¿-Functions
..................... 133
10.4
Primes in Arithmetic Progressions
.............. 134
10.5
Supplementary Problems
.................... 136
11
Density Theorems
. 139
11.1
Counting Ideals in a Fixed Ideal Class
............ 139
11.2
Distribution of Prime Ideals
.................. 146
11.3
The Chebotarev density theorem
............... 150
11.4
Supplementary Problems
.................... 153
CONTENTS xv
II
Solutions
1
Elementary Number Theory
159
1.1
Integers
............................. 159
1.2
Applications of Unique Factorization
............. 166
1.3
The ABC Conjecture
..................... 170
1.4
Supplementary Problems
.................... 173
2
Euclidean Rings
179
2.1
Preliminaries
.......................... 179
2.2
Gaussian Integers
........................ 181
2.3 Eisenstein
Integers
....................... 185
2.4
Some Further Examples
.................... 187
2.5
Supplementary Problems
.................... 189
3
Algebraic Numbers and Integers
197
3.1
Basic Concepts
......................... 197
3.2
Liouville s Theorem and Generalizations
........... 198
3.3
Algebraic Number Fields
.................... 199
3.4
Supplementary Problems
.................... 202
4
Integral Bases
207
4.1
The Norm and the Trace
.................... 207
4.2
Existence of an Integral Basis
................. 208
4.3
Examples
............................ 211
4.4
Ideals in 0K
........................... 213
4.5
Supplementary Problems
.................... 214
5
Dedekind Domains
227
5.1
Integral Closure
......................... 227
5.2
Characterizing Dedekind Domains
.............. 228
5.3
Fractional Ideals and Unique Factorization
.......... 229
5.4
Dedekind s Theorem
...................... 233
5.5
Factorization in
Οχ
...................... 234
5.6
Supplementary Problems
.................... 235
6
The Ideal Class Group
245
6.1
Elementary Results
....................... 245
6.2
Finiteness of the Ideal Class Group
.............. 245
6.3
Diophantine Equations
..................... 247
6.4
Exponents of Ideal Class Groups
............... 248
6.5
Supplementary Problems
.................... 250
xvi CONTENTS
7
Quadratic Reciprocity
263
7.1
Preliminaries
.......................... 263
7.2
Gauss Sums
........................... 266
7.3
The Law of Quadratic Reciprocity
.............. 267
7.4
Quadratic Fields
........................ 270
7.5
Primes in Special Progressions
................ 270
7.6
Supplementary Problems
.................... 272
8
The Structure of Units
279
8.1
Dirichlet s Unit Theorem
................... 279
8.2
Units in Real Quadratic Fields
................ 284
8.3
Supplementary Problems
.................... 291
9
Higher Reciprocity Laws
299
9.1
Cubic Reciprocity
....................... 299
9.2 Eisenstein
Reciprocity
..................... 303
9.3
Supplementary Problems
.................... 308
10
Analytic Methods
313
10.1
The Riemann and Dedekind
Zeta
Functions
......... 313
10.2
Zeta
Functions of Quadratic Fields
.............. 316
10.3
Dirichlet s ¿-Functions
..................... 320
10.4
Primes in Arithmetic Progressions
.............. 322
10.5
Supplementary Problems
.................... 324
11
Density Theorems
333
11.1
Counting Ideals in a Fixed Ideal Class
............ 333
11.2
Distribution of Prime Ideals
.................. 337
11.3
The Chebotarev density theorem
............... 340
11.4
Supplementary Problems
.................... 341
Bibliography
347
Index
349
|
| any_adam_object | 1 |
| author | Murty, Maruti Ram 1953- Esmonde, Jody |
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| dewey-full | 512.7/4 |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7/4 |
| dewey-search | 512.7/4 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| spelling | Murty, Maruti Ram 1953- Verfasser (DE-588)120837307 aut Problems in algebraic number theory M. Ram Murty ; Jody Esmonde 2. ed. New York, NY Springer 2005 XVI, 352 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 190 Nombres algébriques, Théorie des - Problèmes et exercices Algebraic number theory Problems, exercises, etc Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf 1\p (DE-588)4143389-0 Aufgabensammlung gnd-content Analytische Zahlentheorie (DE-588)4001870-2 s DE-604 Algebraische Zahlentheorie (DE-588)4001170-7 s 2\p DE-604 Esmonde, Jody Verfasser aut Graduate texts in mathematics 190 (DE-604)BV000000067 190 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012967898&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Murty, Maruti Ram 1953- Esmonde, Jody Problems in algebraic number theory Graduate texts in mathematics Nombres algébriques, Théorie des - Problèmes et exercices Algebraic number theory Problems, exercises, etc Analytische Zahlentheorie (DE-588)4001870-2 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| subject_GND | (DE-588)4001870-2 (DE-588)4001170-7 (DE-588)4143389-0 |
| title | Problems in algebraic number theory |
| title_auth | Problems in algebraic number theory |
| title_exact_search | Problems in algebraic number theory |
| title_full | Problems in algebraic number theory M. Ram Murty ; Jody Esmonde |
| title_fullStr | Problems in algebraic number theory M. Ram Murty ; Jody Esmonde |
| title_full_unstemmed | Problems in algebraic number theory M. Ram Murty ; Jody Esmonde |
| title_short | Problems in algebraic number theory |
| title_sort | problems in algebraic number theory |
| topic | Nombres algébriques, Théorie des - Problèmes et exercices Algebraic number theory Problems, exercises, etc Analytische Zahlentheorie (DE-588)4001870-2 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| topic_facet | Nombres algébriques, Théorie des - Problèmes et exercices Algebraic number theory Problems, exercises, etc Analytische Zahlentheorie Algebraische Zahlentheorie Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012967898&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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