Problems in algebraic number theory:
Gespeichert in:
| Format: | Elektronisch E-Book |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer
2005
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
190 |
| Schlagworte: | |
| Online-Zugang: | BTU01 TUM01 UBA01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Beschreibung: | 1 Online-Ressource (XVI, 352 S.) |
| ISBN: | 0387221824 9780387269986 |
| DOI: | 10.1007/b138452 |
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Datensatz im Suchindex
| _version_ | 1804136398579564544 |
|---|---|
| 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 0K 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 Dedekind s Theorem 63
5.5 Factorization in Ok 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 L 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 Oa 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 L 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
|
| adam_txt |
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 0K 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 Dedekind's Theorem 63
5.5 Factorization in Ok 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 L 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 Oa' 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 L 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 |
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| series2 | Graduate texts in mathematics |
| spelling | Problems in algebraic number theory M. Ram Murty ; Jody Esmonde 2. ed. New York, NY Springer 2005 1 Online-Ressource (XVI, 352 S.) txt rdacontent c rdamedia cr rdacarrier Graduate texts in mathematics 190 Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 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 Murty, Maruti Ram 1953- Sonstige (DE-588)120837307 oth Esmonde, Jody Sonstige oth Graduate texts in mathematics 190 (DE-604)BV035421258 190 https://doi.org/10.1007/b138452 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015562760&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 | Problems in algebraic number theory Graduate texts in mathematics Algebraische Zahlentheorie (DE-588)4001170-7 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4001170-7 (DE-588)4001870-2 (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_exact_search_txtP | 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 | Algebraische Zahlentheorie (DE-588)4001170-7 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Algebraische Zahlentheorie Analytische Zahlentheorie Aufgabensammlung |
| url | https://doi.org/10.1007/b138452 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015562760&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035421258 |
| work_keys_str_mv | AT murtymarutiram problemsinalgebraicnumbertheory AT esmondejody problemsinalgebraicnumbertheory |