**Graduate Studies in Mathematics**

Volume: 24;
2000;
368 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-2054-4

Product Code: GSM/24

List Price: $72.00

AMS Member Price: $57.60

MAA Member Price: $64.80

**Electronic ISBN: 978-1-4704-2079-6
Product Code: GSM/24.E**

List Price: $72.00

AMS Member Price: $57.60

MAA Member Price: $64.80

# Number Theory: Algebraic Numbers and Functions

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*Helmut Koch*

Algebraic number theory is one of the most refined creations in
mathematics. It has been developed by some of the leading
mathematicians of this and previous centuries. The primary goal of
this book is to present the essential elements of algebraic number
theory, including the theory of normal extensions up through a glimpse
of class field theory. Following the example set for us by Kronecker,
Weber, Hilbert and Artin, algebraic functions are handled here on an
equal footing with algebraic numbers. This is done on the one hand to
demonstrate the analogy between number fields and function fields,
which is especially clear in the case where the ground field is a
finite field. On the other hand, in this way one obtains an
introduction to the theory of “higher congruences” as an
important element of “arithmetic geometry”.

Early chapters discuss topics in elementary number theory, such as
Minkowski's geometry of numbers, public-key cryptography and a short
proof of the Prime Number Theorem, following Newman and Zagier. Next,
some of the tools of algebraic number theory are introduced, such as
ideals, discriminants and valuations. These results are then applied
to obtain results about function fields, including a proof of the
Riemann-Roch Theorem and, as an application of cyclotomic fields, a
proof of the first case of Fermat's Last Theorem. There are a
detailed exposition of the theory of Hecke \(L\)-series,
following Tate, and explicit applications to number theory, such as
the Generalized Riemann Hypothesis. Chapter 9 brings together the
earlier material through the study of quadratic number fields.
Finally, Chapter 10 gives an introduction to class field theory.

The book attempts as much as possible to give simple proofs. It can be
used by a beginner in algebraic number theory who wishes to see some of the
true power and depth of the subject. The book is suitable for two
one-semester courses, with the first four chapters serving to develop the
basic material. Chapters 6 through 9 could be used on their own as a
second semester course.

#### Readership

Mathematicians working in the fields of algebraic number theory and Zeta and \(L\)-functions.

#### Reviews & Endorsements

This book is very suitable for researchers and graduate students who are interested in basic algebraic number theory up to the beginning of class field theory.

-- Mathematica Bohemica

A significant amount of the material goes beyond what one would expect from an introductory text … this book can be recommended to students of number theory for its rigour and emphasis on theory.

-- European Mathematical Society Newsletter

#### Table of Contents

# Table of Contents

## Number Theory: Algebraic Numbers and Functions

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xiii14 free
- Notation xvii18 free
- Chapter 1. Introduction 120 free
- Chapter 2. The Geometry of Numbers 3554
- 2.1. Binary Quadratic Forms 3554
- 2.2. Complete Decomposable Forms of Degree n 3756
- 2.3. Modules and Orders 3958
- 2.4. Complete Modules in Finite Extensions of P 4362
- 2.5. The Integers of a Quadratic Field 4564
- 2.6. Further Examples of Determining a Z-Basis for the Ring of Integers of a Number Field 4665
- 2.7. The Finiteness of the Class Number 4766
- 2.8. The Group of Units 4867
- 2.9. The Start of the Proof of Dirichlet's Unit Theorem 5069
- 2.10. The Rank of 1 (E) 5170
- 2.11. The Regulator of an Order 5574
- 2.12. The Lattice Point Theorem 5574
- 2.13. Minkowski's Geometry of Numbers 5776
- 2.14. Application to Complete Decomposable Forms 6281
- 2.15. Exercises 6483

- Chapter 3. Dedekind's Theory of Ideals 6584
- 3.1. Basic Definitions 6685
- 3.2. The Main Theorem of Dedekind's Theory of Ideals 6887
- 3.3. Consequences of the Main Theorem 7190
- 3.4. The Converse of the Main Theorem 7392
- 3.5. The Norm of an Ideal 7493
- 3.6. Congruences 7695
- 3.7. Localization 7897
- 3.8. The Decomposition of a Prime Ideal in a Finite Separable Extension 8099
- 3.9. The Class Group of an Algebraic Number Field 84103
- 3.10. Relative Extensions 88107
- 3.11. Geometric Interpretation 93112
- 3.12. Different and Discriminant 94113
- 3.13. Exercises 101120

- Chapter 4. Valuations 103122
- 4.1. Fields with Valuation 104123
- 4.2. Valuations of the Field of Rational Numbers and of a Field of Rational Functions 110129
- 4.3. Completion 112131
- 4.4. Complete Fields with Respect to a Discrete Valuation 114133
- 4.5. Extension of a Valuation of a Complete Field to a Finite Extension 121140
- 4.6. Finite Extensions of a Complete Field with a Discrete Valuation 124143
- 4.7. Complete Fields with a Discrete Valuation and Finite Residue Class Field 129148
- 4.8. Extension of the Valuation of an Arbitrary Field to a Finite Extension 132151
- 4.9. Arithmetic in the Compositum of Two Field Extensions 137156
- 4.10. Exercises 137156

- Chapter 5. Algebraic Functions of One Variable 141160
- 5.1. Algebraic Function Fields 142161
- 5.2. The Places of an Algebraic Function Field 144163
- 5.3. The Function Space Associated to a Divisor 149168
- 5.4. Differentials 154173
- 5.5. Extensions of the Field of Constants 158177
- 5.6. The Riemann-Roch Theorem 160179
- 5.7. Function Fields of Genus 0 164183
- 5.8. Function Fields of Genus 1 167186
- 5.9. Exercises 169188

- Chapter 6. Normal Extensions 171190
- 6.1. Decomposition Group and Ramification Groups 172191
- 6.2. A New Proof of Dedekind's Theorem on the Different 176195
- 6.3. Decomposition of Prime Ideals in an Intermediate Field 178197
- 6.4. Cyclotomic Fields 180199
- 6.5. The First Case of Fermat's Last Theorem 184203
- 6.6. Localization 188207
- 6.7. Upper Numeration of the Ramification Group 190209
- 6.8. Kummer Extensions 195214
- 6.9. Exercises 199218

- Chapter 7. L-Series 203222
- 7.1. From the Riemann ζ-Function to the Hecke L-Series 204223
- 7.2. Normalized Valuations 207226
- 7.3. Adeles 209228
- 7.4. Ideles 212231
- 7.5. Idele Class Group and Ray Class Group 214233
- 7.6. Hecke Characters 217236
- 7.7. Analysis on Local Additive Groups 219238
- 7.8. Analysis on the Adele Group 223242
- 7.9. The Multiplicative Group of a Local Field 227246
- 7.10. The Local Functional Equation 230249
- 7.11. Calculation of p(c) for K = R 232251
- 7.12. Calculation of p(c) for K = C 234253
- 7.13. Computation of the p-Factors for a Nonarchimedean Field 236255
- 7.14. Relations Among the p-Factors 239258
- 7.15. Analysis on the Idele Group 240259
- 7.16. Global Zeta Functions 243262
- 7.17. The Dedekind Zeta Function 247266
- 7.18. Hecke L-Series 251270
- 7.19. Congruence Zeta Functions 252271
- 7.20. Exercises 257276

- Chapter 8. Applications of Hecke L-Series 259278
- Chapter 9. Quadratic Number Fields 275294
- 9.1. Quadratic Forms and Orders in Quadratic Number Fields 275294
- 9.2. The Class Number of Imaginary Quadratic Number Fields 282301
- 9.3. Continued Fractions 285304
- 9.4. Periodic Continued Fractions 290309
- 9.5. The Fundamental Unit of an Order of a Real Quadratic Number Field 295314
- 9.6. The Character of a Quadratic Number Field 301320
- 9.7. The Arithmetic Class Number Formula 303322
- 9.8. Computing the Gaussian Sum 310329
- 9.9. Exercises 313332

- Chapter 10. What Next? 315334
- Appendix A. Divisibility Theory 325344
- Appendix B. Trace, Norm, Different, and Discriminant 341360
- Appendix C. Harmonic Analysis on Locally Compact Abelian Groups 345364
- References 359378
- Index 363382
- Back Cover Back Cover1388