GALOIS-GROUPS AND THE MULTIPLICATIVE STRUCTURE OF FIELD-EXTENSIONS

Let K/k be a finite Galois field extension, and assume k is not an algebraic extension of a finite field. Let K* 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 */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.