Kummer Theories for Algebraic Tori and Normal Basis Problem

Abstract

We discuss the inverse Galois problem with normal basis, concerning Kummer theories for algebraic tori, in the framework of group schemes. The unit group scheme of a group algebra plays an important role in this article, as was pointed out by Serre~[8]. We develop our argument not only over a field but also over a ring, considering integral models of Kummer theories for algebraic tori.