Algebraic Number Theory

Abstract

This book originates from graduate courses given in Cambridge and London. It provides a brisk, thorough treatment of the foundations of algebraic number theory, and builds on that to introduce more advanced ideas. Throughout, the authors emphasise the systematic development of techniques for the explicit calculation of the basic invariants, such as rings of integers, class groups, and units. Moreover they combine, at each stage of development, theory with explicit computations and applications, and provide motivation in terms of classical number-theoretic problems. A number of special topics are included that can be treated at this level but can usually only be found in research monographs or original papers, for instance: module theory of Dedekind domains; tame and wild ramifications; Gauss series and Gauss periods; binary quadratic forms; and Brauer relations. This is the only textbook at this level which combines clean, modern algebraic techniques together with a substantial arithmetic content. It will be indispensable for all practising and would-be algebraic number theorists.

Weight distributions of geometric Goppa codes

“…When written out they express the first and second orthogonality relations on characters. A direct proof of the first claim is in [10,Appendix].…”

“…The main theorems of class field theory go far beyond what is needed to prove (10). We recall briefly that only (part of) Hilbert's theorem on class fields is needed.…”

“…We will see that the proper interpretation of Hilbert's theorem for K a function field will imply (10). A careful translation is carried out in [23].…”

Congruence covers of triangular modular curves and their Galois groups

“…Our goal in this section is to state some basic facts from algebraic number theory and give an explicit formula for f (B + |p) that will be used later. Unless otherwise mentioned, the facts and definitions in this section can be found in most algebraic number theory textbooks like [Mar77] and [FT93].…”

Sur les corps de Hilbert-Speiser

“…= a+b √ p, avec a pair). On se servira de la caractérisation de l'unité fondamentale suivante (voir [2], proposition 1.6, p.178) :…”