An application of localization to Galois cohomology

Abstract

We use a localization theorem and a characterization of the first group of cohomology [H.sup.1](G, B) to give a new proof that the groups of cohomology [H.sup.i] (G, B) of finite cyclic extensions of number fields have same order for all integers i. This result was proved by H. Yokoy in [10] by using the theorem on existence of a normal basis.