Abstract
LetK/kbe an extension of degreep2over a p-adic number fieldkwith the Galois groupG. We study the Galois module structure of the ring OKof integers inK. We determine conditions under which the invariant factors of Kummer orders OKin OKof two extensions coincide with each other and give two examples, one of which shows there exist Kummer extensionsKandLwithD(K)=D(L) such that OKand OLare not ZpG-isomorphic. The other shows the existence of extensionsFandKsuch that OFand OKare isomorphic over ZpGbut not over okG.
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