Galois theoretic local-global relations in nilpotent extensions of algebraic number fields

Abstract

The Galois group of the maximal abelian extension of an algebraic number field of finite degree is excellently described by class field theory with ideles of the ground field, which is due to C. Chevalley. We see by it how the local arithmetic phenomena are tied up as a glogal whole by the relations determined by the global numbers. The purpose of this note is to give an “analogous” description of the Galois group of the maximal nilpotent extensions of the ground field which may be regarded as a “Galois theoretic lifting” of the abelian case.