An Invitation To Algebraic Numbers And Algebraic Functions: 1st Edition (Hardback) book cover

An Invitation To Algebraic Numbers And Algebraic Functions

1st Edition

By Franz Halter-Koch

Chapman and Hall/CRC

568 pages | 3 B/W Illus.

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Hardback: 9781138583610
pub: 2020-05-13
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Description

The author offers a thorough presentation of the classical theory of algebraic numbers and algebraic functions which both in its conception and in many details differs from the current literature on the subject. The basic features are: Field-theoretic preliminaries and a detailed presentation of Dedekind’s ideal theory including non-principal orders and various types of class groups; the classical theory of algebraic number fields with a focus on quadratic, cubic and cyclotomic fields; basics of the analytic theory including the prime ideal theorem, density results and the determination of the arithmetic by the class group; a thorough presentation of valuation theory including the theory of differents, discriminants, and higher ramification. The theory of function fields is based on the ideal and valuation theory developed before; it presents the Riemann-Roch theorem on the basis of Weil differentials and highlights in detail the connection with classical differentials. The theory of congruence zeta functions and a proof of the Hasse -Weil theorem represent the culminating point of the volume.

The volume is accessible with a basic knowledge in algebra and elementary number theory. It empowers the reader to follow the advanced number-theoretic literature, and it is a solid basis for the study of the forthcoming volume on the foundations and main results of class field theory.

Key features:

  • A thorough presentation of the theory of Algebraic Numbers and Algebraic Functions on an ideal- and valuation-theoretic basis.
  • Several of the topics both in the number field and in the function field case were not presented before in this context.
  • Despite the wealth of presented advanced topics the text is easily readable.

Table of Contents

Topological groups and infinite Galois theory. Cohomology of groups. Simple algebras. Local class field theory. Idels and holomorphy domains in global fields. Global class field theory. Artin L functions

About the Author

Franz Halter-Koch studied at Universities of Graz and Hamburg under Helmut Hasse and Alexander Aigner. He has been an Assistant Professor at University of Cologne, and a Full Professor at University of Essen and University of Graz. He has 156 research articles published in various journals. His books include Ideal Systems (Marcel Dekker/CRC Press); Non-Unique Factorizations (Chapman&Hall/CRC), and Quadratic Irrationals, (Chapman&Hall/CRC).

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT002000
MATHEMATICS / Algebra / General
MAT022000
MATHEMATICS / Number Theory