Über relativ-lnvariante Kreiseinheiten und Stickelberger-Elemente

We consider an extensionE ?K of absolutely abellan number fields and the corresponding groups of circular unitsC _E ?C _K in the sense of Sinnott. In this paper we consider the question: Is every Gal(K/E)-invarlant element ofC _K already inC _E ? This has been answered in the affirmative recently by Gold and Kirn in the case that bothE andK are cyclotomlc fields. We show that the question has an affirmative answer ifK is cyclic over ?, and that the answer in general is negative. There is an analogous question concerning Stickelberger ideals (the inclusion map now being replaced by the corestriction map), and the answer to that question is shown to be exactly the same as to the first one.