Introduction to the theory of algebraic numbers and functions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
New York <[u.a.]>
Acad. Pr.
1966
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| Series: | Pure and applied mathematics
23 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV,324 S. |
Staff View
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| 240 | 1 | 0 | |a Einführung in die Theorie der algebraischen Zahlen und Funktionen |
| 245 | 1 | 0 | |a Introduction to the theory of algebraic numbers and functions |
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Record in the Search Index
| _version_ | 1804117261722583040 |
|---|---|
| adam_text | Contents
Preface to the English Edition v
Preface to the German Edition vii
Introduction
1. The Subject 1
2. The Method 2
Table of Several Abbreviations and Symbols 3
Chapter I: Linear Algebra
§1. Modules in Principal Ideal Domains 5
1. Finite Modules .5
2. The Theorem of Elementary Divisors 7
3. Dual Spaces and Complementary Modules 10
4.* Noetherian Rings 11
5.* A Further Basis Theorem 13
§2. Systems of Linear Inequalities 14
1. Minkowski s Point Lattice Theorem 14
2.* Siegel s Proof 16
3. Generalization to Function Fields 18
§3. Linear Divisors 20
1. Basic Concepts 20
2. Norm and Degree of a Linear Divisor 23
3. The Dimension of a Linear Divisor 23
4.* The Riemann Roch Theorem and the Minkowski Linear Form Theorem 26
§4. Traces, Norms, and Discriminants 27
1. Representations by Matrices 27
2. The Transitivity Formulas 28
3. The Discriminant 29
4. Separable and Inseparable Extensions 30
* The sections marked with an asterisk might be omitted in the first reading.
ix
X CONTENTS
Appendix to Chapter I: The Theta Function
§1* The Symplectic Group 32
1. The Basic Properties 32
2. Symplectic Geometry 34
3. The Hyperbolic Plane and Hyperbolic Space 35
4. The Symplectic Modular Group 36
5. The Fundamental Domain 39
6. The Theta Function 41
7. Proof of the Reciprocity Formula 43
§2.* Theta Functions for Quadratic Forms 44
1. Simple Gaussian Sums 44
2. The Quadratic Reciprocity Law and the Sign of Gaussian Sums 46
3. The Theta Function of a Definite Quadratic Form 48
Chapter II: Ideals and Divisors
§1. Ideals 53
1. Integral Dependence 53
2. The Finiteness of the Principal Order 54
3. Kronecker Divisors 56
4. Ideals 58
5. Proof of the Principal Theorem 60
6.* Extension of Divisibility Theory 61
§2. Local Rings 63
1. Basic concepts 63
2. Local Rings in Algebraic Extensions 64
3. Local Rings in Algebraic Number and Function Fields 66
4. The Component Decomposition of Ideals 67
§3. Ideals in Different Fields; the Norm 70
1. Extension of an Ideal 70
2. The Norm 70
3. The Prime Ideals 72
§4. The Complement, Different, and Discriminant 73
1. The Complement 73
2. Different and Discriminant 75
3. The Dedekind Discriminant Theorem 77
§5. Divisors 79
1. The Rational Functici Field 79
2.* Projective Invariance 81
CONTENTS Xi
3. Divisors in Algebraic Number and Function Fields 82
4. The Behavior of Divisors under Field Extensions 84
5. The Prime Divisors 86
6. Divisors and Linear Divisors 88
7. The Linear Degree 89
§6.* Decomposition of Prime Ideals in Galois Extensions 90
1. The Decomposition Group and Inertia Group 91
2. The Ramification Groups 94
3. The Discriminant 96
Appendix to Chapter II:* Topics from the Theory of Algebraic Number
Fields
§1. The Finiteness Theorems 98
1. The Finiteness of the Ideal Class Number 98
2. The Discriminant 100
3. The Dirichlet Unit Theorem 101
4. The Regulator 104
§2. Quadratic Number Fields and Cyclotomic Fields 104
1. Quadratic Number Fields 104
2. Special Cyclotomic Fields 106
Chapter III: Algebraic Functions and Differentials
§1. Power Series Expansions of Algebraic Functions 110
1. The Field of Power Series 110
2. Divisibility, Rearranging of Power Series 112
3. Inversion of a Power Series 113
4. Algebraic Functions; Regular Places 115
5. Continuation; Critical Places 116
6. Puiseux s Theorem 118
§2. Algebraic Function Fields 120
1. Divisors in Rational Function Fields 120
2. Divisors in Algebraic Function Fields 121
3. Decomposition of Rational Divisors 124
4. The Principal Orders 125
5. Divisors and Linear Divisors 128
6. The Invariance of the Concept of Divisors 131
7.* Extension to More General Constant Fields 131
§3. The Riemann Roch Theorem 132
1. Dimension of a Divisor Class 132
2. The Riemann Roch Theorem 133
Xii CONTENTS
3. Questions of Invariance 135
4. Extension of the Constant Field 135
5. The Fields of Genus 0 139
6. Luroth s Theorem 140
7. Further Proofs and Generalizations of the Riemann Roch Theorem 141
References 142
§4. Differentials 143
1. Differential Quotients 143
2. The Differential Calculus with Characteristic p 144
3. The Concept of the Differential 147
4.* Continuation; Separable and Inseparable Prime Divisors 149
5. Cartier s Operator 150
6. Residues of Differentials 151
7. The Residue Theorem 153
8.* The Differential Class 156
§5. Differentials and Principal Part Systems 158
1. Differentials of Higher Degrees 158
2. Principal Part Systems 15£
3. The Scalar Product 160
4. The Relationship to Integral Calculus 163
5.* The Diagonal 165
6.* The Analog of the Green Function 167
Notes 171
§6. Reduction of a Function Field with Respect to a Prime Ideal
of the Constant Field 171
1. The Irreducibility Theorem 171
2. Regular Prime Ideals 174
3. Behavior of Ideals under Residue Formation 177
4. Behavior of Divisors under Residue Formation 178
5. Continuation; Behavior of Differentials under Residue Formation 181
6. Behavior of the Field under Residue Formation and Extension 183
Notes 183
References 184
Chapter IV: Algebraic Functions over the Complex Number Field
§1. Riemann Surfaces 185
1. The Riemann Surface of an Algebraic Function 185
2. The Riemann Surface as Complex Manifold 186
3. The Riemann Surface as Topological Manifold 188
§2. Fields of Elliptic Functions 190
1. Introduction 190
2. The Addition Theorem 191
3. Automorphisms 193
CONTENTS xiii
4. The Integral of the First Kind 196
5. The Addition Theorem and the Abel Theorem 197
6. The Weierstrass Normal Form 200
7. Elementary Elliptic Functions 202
Notes 203
References 204
§3. The Group of Divisor Classes of Degree 0 204
1. The Riemann Period Matrix 204
2. A Hermitian Metric for Differentials of the First Kind 206
3. Abelian Integrals of the Third Kind 208
4. Abel s Theorem 209
5. The Jacobian Variety 211
Notes 213
References 214
§4. Modular Functions 214
1. The Modular Surface 214
2. Covering Spaces of the Modular Surface 215
3. Congruence Subgroups 217
4. Modular Forms 219
5. The Field of Modular Functions 221
6. Modular Forms and Differentials 223
7. Fourier Expansions of Eisenstein Series 225
8. Theta Functions 229
References 232
Chapter V: Correspondences between Fields of Algebraic Functions
§1. The Correspondences 233
1. Basic Concepts 233
2. Multiplication of Correspondences 236
3. Properties of the Product 239
4. Correspondences of a Field with Itself 241
5. Effect of Correspondences on Divisors 242
6. Prime Correspondences 245
7. Inseparable Extensions 246
8. The Frobenius Automorphism 249
9. Correspondences of a Field of Automorphic Functions with Itself 250
§2. Representations of Correspondences in the Space of Differentials 252
1. Definitions 252
2. The Classical Case 256
3. Continuation; Representations of Rosati Adjoint Correspondences 258
4. The Trace 259
5. Evaluation of the Trace Formula 263
Notes 265
Xiv CONTENTS
§3. Modular Functions 266
1. The Modular Correspondences 266
2. Products of Modular Correspondences 269
3. Representations of Modular Correspondences by Differentials 271
4. The Petersson Metric 272
5. Fourier Expansions of Modular Forms 275
6. Ramanujan s Conjecture 277
7. Results for Modular Forms of Odd Dimensions; Notes 279
References 281
§4. Castelnuovo s Inequality 281
1. Introduction 281
2. Reduction to the Classical Case 282
3. Extension of the Notion of Correspondence 284
4. The Fixed Points of a Correspondence 286
5. The Connection with §2 289
6. The Trace 291
7. Second Proof of the Principal Theorem: Preparations 293
8. Second Proof of the Principal Theorem: Conclusion 295
9.* Remarks Concerning the Ring of Correspondence Classes 297
Notes 298
References 299
§5. Applications in Number Theory 299
1. The Zeta Function of a Field of Functions 299
2. The Functional Equation 301
3. Extension of the Field of Constants 303
4. Riemann s Conjecture 305
5. Modular Functions 307
6. The Eigenvalues of Modular Correspondences 311
7. Modular Functions of the Principal Character 313
Notes 314
References 315
§6. Elliptic Function Fields 315
1. The Ring of Correspondence Classes 316
2. Complex Multiplication 318
Author Index 321
Subject Index 322
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| spelling | Eichler, Martin Verfasser aut Einführung in die Theorie der algebraischen Zahlen und Funktionen Introduction to the theory of algebraic numbers and functions New York <[u.a.]> Acad. Pr. 1966 XIV,324 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 23 Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Algebraische Funktion (DE-588)4141836-0 gnd rswk-swf Algebraische Zahl (DE-588)4141847-5 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Theorie (DE-588)4059787-8 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 s DE-604 Algebraische Funktion (DE-588)4141836-0 s Theorie (DE-588)4059787-8 s Algebraische Zahl (DE-588)4141847-5 s 1\p DE-604 Zahlentheorie (DE-588)4067277-3 s 2\p DE-604 Pure and applied mathematics 23 (DE-604)BV010177228 23 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001859889&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 | Eichler, Martin Introduction to the theory of algebraic numbers and functions Pure and applied mathematics Zahlentheorie (DE-588)4067277-3 gnd Algebraische Funktion (DE-588)4141836-0 gnd Algebraische Zahl (DE-588)4141847-5 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Theorie (DE-588)4059787-8 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4141836-0 (DE-588)4141847-5 (DE-588)4001170-7 (DE-588)4059787-8 |
| title | Introduction to the theory of algebraic numbers and functions |
| title_alt | Einführung in die Theorie der algebraischen Zahlen und Funktionen |
| title_auth | Introduction to the theory of algebraic numbers and functions |
| title_exact_search | Introduction to the theory of algebraic numbers and functions |
| title_full | Introduction to the theory of algebraic numbers and functions |
| title_fullStr | Introduction to the theory of algebraic numbers and functions |
| title_full_unstemmed | Introduction to the theory of algebraic numbers and functions |
| title_short | Introduction to the theory of algebraic numbers and functions |
| title_sort | introduction to the theory of algebraic numbers and functions |
| topic | Zahlentheorie (DE-588)4067277-3 gnd Algebraische Funktion (DE-588)4141836-0 gnd Algebraische Zahl (DE-588)4141847-5 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Theorie (DE-588)4059787-8 gnd |
| topic_facet | Zahlentheorie Algebraische Funktion Algebraische Zahl Algebraische Zahlentheorie Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001859889&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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