Chapman and Hall/CRC
568 pages | 3 B/W Illus.
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.
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