Class Field Theory

Frontmatter

Introduction to Global Class Field Theory

Georges Gras

I. Basic Tools and Notations

Abstract

This chapter gives the definitions of the objects which will be used throughout this book. We are thus led to give the main general notations.

Georges Gras

II. Reciprocity Maps Existence Theorems

Abstract

The fundamental results given in this chapter do not necessarily form a sequence of logical steps for a proof of class field theory, but are written and commented so as to be used. This is so true that, as we will see several times, a classical proof consists in deducing local class field theory from global class field theory, as was initiated by Hasse and Schmidt in 1930, and in particular to base some local computations on global arguments (a typical example being the global computation of a local Hilbert symbol in 7.5); however here, in the description of the results, we will go from local to global, which seems more natural.

Georges Gras

III. Abelian Extensions with Restricted Ramification — Abelian Closure

Abstract

This chapter deals with the correspondence of class field theory both for finite and infinite extensions; this second aspect, obtained by limiting processes, will enable us to understand the structure of the maximal abelian extension of a number field K (Section 4 of the present chapter). Indeed, since any finite abelian extension of K is contained in a ray class field K(m)^res, we have \({\overline K ^{ab}}\, = \,\mathop U\limits_m \,K{(m)^{res}}\), where m ranges in the set of moduli of K.

Georges Gras

IV. Invariant Class Groups in p-Ramification Genus Theory

Abstract

If the arithmetical invariants of K are known, in other words if class field theory over K is explicit, the situation for a finite extension L is a priori completely different, and one usually studies the corresponding invariants of L using several means. This chapter explains the two classical approaches: invariant classes formulas and genus theory.

Georges Gras

V. Cyclic Extensions with Prescribed Ramification

Abstract

In this chapter we give an approach to the study of ramification in \({\bar K^{ab}}\left[ {{p^e}} \right]/K\), the maximal pro-p-subextension of\({\bar K^{ab}}/K\) with exponent p ^e , in particular through the study of the ramification possibilities for cyclic extensions of degree p ^e of K. We will apply these results to the case of the maximal tamely ramified abelian extension \(H_{ta}^{res}/K\) whose structure is always complicated as soon as the invariants \({{\rm A}^{res}}\) or E ^res are nontrivial. Concerning this, we will have to make an assumption on the group \({\left( {{{\rm A}^{res}}} \right)_p}\) when e ≥ 2, but the case e = 1 can be solved without any assumption.

Georges Gras

Erratum

Georges Gras

Backmatter