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 difference, 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 is a solid basis for the study of the forthcoming volume on the foundations and main results of class field theory.
• 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 presenting many advanced topics, the text is easily readable.
Franz Halter-Koch is professor emeritus at the university of Graz. He is the author of “Ideal Systems” (Marcel Dekker,1998), “Quadratic Irrationals” (CRC, 2013), and a co-author of “Non-Unique Factorizations” (CRC 2006).
Table of Contents
1 Field Extensions
2 Dedekind Theory
3 Algebraic Number Fields: Elementary and Geometric Methods
4 Elementary Analytic Theory
5 Valuation Theory
6 Algebraic Function Fields
List of Symbols
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).
"...Koch is extremely thorough, very incisive, and very careful --- all great pedagogical virtues, present in spades. He arranges his results very well, phrasing things carefully and explicitly, and his proofs are detailed. I tend to cover the margins of the books I read with everything from disputes and questions to proof sketches. Koch’s book would require only a minimum of this sort of polemics: it’s all there --- no guesswork. The additional blood, sweat, and tears attending learning mathematics well, i.e. doing problems, problems, problems, is represented by 20 problems attached to each of Koch’s six chapters. Scanning them, they look excellent to me: they should serve the reader very well indeed. And that’s true for the entire book: it’s excellent and is well worth using in order to learn this beautiful material. I look forward to Koch’s book of class field theory!"
- Michael Berg, Loyola Marymount University, Published in MAA