Number theory : algebraic numbers and functions / Helmut Koch ; translated by David Kramer.

Format:

Book

Author/Creator:

Koch, Helmut, 1932-

Contributor:

Kramer, David, translator.

Series:

Graduate studies in mathematics ; v. 24.

Graduate studies in mathematics, 1065-7339 ; volume 24

Standardized Title:

Zahlentheorie. English

Language:

English

Subjects (All):

Number theory.

Physical Description:

1 online resource (388 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Language Note:

English

Summary:

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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Translator's Note""; ""Notation""; ""List of Symbols""; ""Chapter 1. Introduction""; ""1.1. Pythagorean Triples""; ""1.2. Pell's Equation""; ""1.3. Fermat's Last Theorem""; ""1.4. Congruences""; ""1.5. Public Key Cryptology""; ""1.6. Quadratic Residues""; ""1.7. Prime Numbers""; ""1.8. The Prime Number Theorem""; ""1.9. Exercises""; ""Chapter 2. The Geometry of Numbers""; ""2.1. Binary Quadratic Forms""; ""2.2. Complete Decomposable Forms of Degree n""; ""2.3. Modules and Orders""

""2.4. Complete Modules in Finite Extensions of P""""2.5. The Integers of a Quadratic Field""; ""2.6. Further Examples of Determining a Z-Basis for the Ring of Integers of a Number Field""; ""2.7. The Finiteness of the Class Number""; ""2.8. The Group of Units""; ""2.9. The Start of the Proof of Dirichlet's Unit Theorem""; ""2.10. The Rank of 1 (E)""; ""2.11. The Regulator of an Order""; ""2.12. The Lattice Point Theorem""; ""2.13. Minkowski's Geometry of Numbers""; ""2.14. Application to Complete Decomposable Forms""; ""2.15. Exercises""; ""Chapter 3. Dedekind's Theory of Ideals""

""3.1. Basic Definitions""""3.2. The Main Theorem of Dedekind's Theory of Ideals""; ""3.3. Consequences of the Main Theorem""; ""3.4. The Converse of the Main Theorem""; ""3.5. The Norm of an Ideal""; ""3.6. Congruences""; ""3.7. Localization""; ""3.8. The Decomposition of a Prime Ideal in a Finite Separable Extension""; ""3.9. The Class Group of an Algebraic Number Field""; ""3.10. Relative Extensions""; ""3.11. Geometric Interpretation""; ""3.12. Different and Discriminant""; ""3.13. Exercises""; ""Chapter 4. Valuations""; ""4.1. Fields with Valuation""

""4.2. Valuations of the Field of Rational Numbers and of a Field of Rational Functions""""4.3. Completion""; ""4.4. Complete Fields with Respect to a Discrete Valuation""; ""4.5. Extension of a Valuation of a Complete Field to a Finite Extension""; ""4.6. Finite Extensions of a Complete Field with a Discrete Valuation""; ""4.7. Complete Fields with a Discrete Valuation and Finite Residue Class Field""; ""4.8. Extension of the Valuation of an Arbitrary Field to a Finite Extension""; ""4.9. Arithmetic in the Compositum of Two Field Extensions""; ""4.10. Exercises""

""Chapter 5. Algebraic Functions of One Variable""""5.1. Algebraic Function Fields""; ""5.2. The Places of an Algebraic Function Field""; ""5.3. The Function Space Associated to a Divisor""; ""5.4. Differentials""; ""5.5. Extensions of the Field of Constants""; ""5.6. The Riemann-Roch Theorem""; ""5.7. Function Fields of Genus 0""; ""5.8. Function Fields of Genus 1""; ""5.9. Exercises""; ""Chapter 6. Normal Extensions""; ""6.1. Decomposition Group and Ramification Groups""; ""6.2. A New Proof of Dedekind's Theorem on the Different""

""6.3. Decomposition of Prime Ideals in an Intermediate Field""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-2079-1