Title: Galois Group of the Maximal Abelian Extension over an Algebraic Number Field
Abstract: The aim of the present work is to determine the Galois group of the maximal abelian extension Ω A over an algebraic number field Ω of finite degree, which we fix once for all. Let Z be a continuous character of the Galois group of Ω A /Ω . Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Z character of Ω . For our purpose, it suffices to determine the group X l of the characters of Ω whose orders are powers of a prime number l .