Galois groups and the multiplicative structure of field extensions

Abstract

Let $K/k$ be a finite Galois field extension, and assume $k$ is not an algebraic extension of a finite field. Let ${K^{\ast } }$ be the multiplicative group of $K$, and let $\Theta (K/k)$ be the product of the multiplicative groups of the proper intermediate fields. The condition that the quotient group $\Gamma = {K^{\ast } }/\Theta (K/k)$ be torsion is shown to depend only on the Galois group $G$. For algebraic number fields and function fields, we give a complete classification of those $G$ for which $\Gamma$ is nontrivial.