Algebraic number theory and Fermat's last theorem:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Natick, Mass.
Peters
2002
|
| Ausgabe: | 3. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Bis 2. Aufl. u.d.T.: Stewart, Ian: Algebraic number theory |
| Beschreibung: | XIX, 313 S. Ill., graph. Darst. |
| ISBN: | 1568811195 |
Internformat
MARC
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| 245 | 1 | 0 | |a Algebraic number theory and Fermat's last theorem |c Ian Stewart ; David Tall |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Natick, Mass. |b Peters |c 2002 | |
| 300 | |a XIX, 313 S. |b Ill., graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Bis 2. Aufl. u.d.T.: Stewart, Ian: Algebraic number theory | ||
| 650 | 4 | |a Fermat, Grand théorème de | |
| 650 | 4 | |a Nombres algébriques, Théorie des | |
| 650 | 7 | |a Números algébricos |2 larpcal | |
| 650 | 7 | |a Teoria dos números |2 larpcal | |
| 650 | 4 | |a Algebraic number theory | |
| 650 | 4 | |a Fermat's last theorem | |
| 650 | 0 | 7 | |a Fermatsche Vermutung |0 (DE-588)4154012-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Algebraische Zahlentheorie |0 (DE-588)4001170-7 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804129201081548800 |
|---|---|
| adam_text | Contents
Preface xi
Index of Notation xvii
The Origins of Algebraic Number Theory 1
I Algebraic Methods 7
1 Algebraic Background 9
1.1 Rings and Fields 10
1.2 Factorization of Polynomials 13
1.3 Field Extensions 20
1.4 Symmetric Polynomials 22
1.5 Modules 25
1.6 Free Abelian Groups 26
1.7 Exercises 32
2 Algebraic Numbers 35
2.1 Algebraic Numbers 36
2.2 Conjugates and Discriminants 38
2.3 Algebraic Integers 42
2.4 Integral Bases 45
2.5 Norms and Traces 49
2.6 Rings of Integers 51
2.7 Exercises 57
vii
viii Contents
3 Quadratic and Cyclotomic Fields 61
3.1 Quadratic Fields 61
3.2 Cyclotomic Fields 64
3.3 Exercises 69
4 Factorization into Irreducibles 73
4.1 Historical Background 75
4.2 Trivial Factorizations 76
4.3 Factorization into Irreducibles 79
4.4 Examples of Non Unique Factorization into Irreducibles . . 82
4.5 Prime Factorization 86
4.6 Euclidean Domains 90
4.7 Euclidean Quadratic Fields 91
4.8 Consequences of Unique Factorization 94
4.9 The Ramanujan Nagell Theorem 96
4.10 Exercises 99
5 Ideals 101
5.1 Historical Background 102
5.2 Prime Factorization of Ideals 105
5.3 The Norm of an Ideal 114
5.4 Nonunique Factorization in Cyclotomic Fields 122
5.5 Exercises 124
n Geometric Methods 127
6 Lattices 129
6.1 Lattices 129
6.2 The Quotient Torus 132
6.3 Exercises 136
7 Minkowski s Theorem 139
7.1 Minkowski s Theorem 139
7.2 The Two Squares Theorem 142
7.3 The Four Squares Theorem 143
7.4 Exercises 144
8 Geometric Representation of Algebraic Numbers 145
8.1 The Space mathbfLst 145
8.2 Exercises 149
Contents ix
9 Class Group and Class Number 151
9.1 The Class Group 152
9.2 An Existence Theorem 153
9.3 Finiteness of the Class Group 157
9.4 How to Make an Ideal Principal 158
9.5 Unique Factorization of Elements in an Extension Ring . . 162
9.6 Exercises 164
III Number Theoretic Applications 167
10 Computational Methods 169
10.1 Factorization of a Rational Prime 169
10.2 Minkowski s Constants 172
10.3 Some Class Number Calculations 176
10.4 Tables 179
10.5 Exercises 180
11 Rummer s Special Case of Fermafs Last Theorem 183
11.1 Some History 183
11.2 Elementary Considerations 186
11.3 Kummer s Lemma 189
11.4 Kummer s Theorem 193
11.5 Regular Primes 196
11.6 Exercises 198
12 The Path to the Final Breakthrough 201
12.1 The Wolfskehl Prize 201
12.2 Other Directions 203
12.3 Modular Functions and Elliptic Curves 205
12.4 The Taniyama Shimura Weil Conjecture 206
12.5 Frey s Elliptic Equation 207
12.6 The Amateur who Became a Model Professional 207
12.7 Technical Hitch 210
12.8 Flash of Inspiration 211
12.9 Exercises 212
13 Elliptic Curves 213
13.1 Review of Conies 214
13.2 Projective Space 215
13.3 Rational Conies and the Pythagorean Equation 220
13.4 Elliptic Curves 222
13.5 The Tangent/Secant Process 225
x Contents
13.6 Group Structure on an Elliptic Curve 226
13.7 Applications to Diophantine Equations 230
13.8 Exercises 233
14 Elliptic Functions 235
14.1 Trigonometry Meets Diophantus 235
14.2 Elliptic Functions 243
14.3 Legendre and Weierstrass 249
14.4 Modular Functions 250
14.5 The Frey Elliptic Curve 256
14.6 The Taniyama Shimura Weil Conjecture 257
14.7 Sketch Proof of Fermat s Last Theorem 261
14.8 Recent Developments 263
14.9 Exercises 268
IV Appendices 271
A Quadratic Residues 273
A.I Quadratic Equations in Zm 274
A.2 The Units of Zm 276
A.3 Quadratic Residues 281
A.4 Exercises 290
B Dirichlet s Units Theorem 293
B.I Introduction 293
B.2 Logarithmic Space 294
B.3 Embedding the Unit Group in Logarithmic Space 295
B.4 Dirichlet s Theorem 296
B.5 Exercises 301
Bibliography 303
Index 309
|
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| indexdate | 2024-07-09T19:01:22Z |
| institution | BVB |
| isbn | 1568811195 |
| language | English |
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| spelling | Stewart, Ian 1945- Verfasser (DE-588)124091598 aut Algebraic number theory and Fermat's last theorem Ian Stewart ; David Tall 3. ed. Natick, Mass. Peters 2002 XIX, 313 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Bis 2. Aufl. u.d.T.: Stewart, Ian: Algebraic number theory Fermat, Grand théorème de Nombres algébriques, Théorie des Números algébricos larpcal Teoria dos números larpcal Algebraic number theory Fermat's last theorem Fermatsche Vermutung (DE-588)4154012-8 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 s Fermatsche Vermutung (DE-588)4154012-8 s 1\p DE-604 Zahlentheorie (DE-588)4067277-3 s 2\p DE-604 Tall, David Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009812640&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 | Stewart, Ian 1945- Tall, David Algebraic number theory and Fermat's last theorem Fermat, Grand théorème de Nombres algébriques, Théorie des Números algébricos larpcal Teoria dos números larpcal Algebraic number theory Fermat's last theorem Fermatsche Vermutung (DE-588)4154012-8 gnd Zahlentheorie (DE-588)4067277-3 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| subject_GND | (DE-588)4154012-8 (DE-588)4067277-3 (DE-588)4001170-7 |
| title | Algebraic number theory and Fermat's last theorem |
| title_auth | Algebraic number theory and Fermat's last theorem |
| title_exact_search | Algebraic number theory and Fermat's last theorem |
| title_full | Algebraic number theory and Fermat's last theorem Ian Stewart ; David Tall |
| title_fullStr | Algebraic number theory and Fermat's last theorem Ian Stewart ; David Tall |
| title_full_unstemmed | Algebraic number theory and Fermat's last theorem Ian Stewart ; David Tall |
| title_short | Algebraic number theory and Fermat's last theorem |
| title_sort | algebraic number theory and fermat s last theorem |
| topic | Fermat, Grand théorème de Nombres algébriques, Théorie des Números algébricos larpcal Teoria dos números larpcal Algebraic number theory Fermat's last theorem Fermatsche Vermutung (DE-588)4154012-8 gnd Zahlentheorie (DE-588)4067277-3 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd |
| topic_facet | Fermat, Grand théorème de Nombres algébriques, Théorie des Números algébricos Teoria dos números Algebraic number theory Fermat's last theorem Fermatsche Vermutung Zahlentheorie Algebraische Zahlentheorie |
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