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.