On the Invariant Factors of Kummer Orders in the Rings of Integers of image-adic Number Fields of Degreep2 Original Research Article

LetK/kbe an extension of degreep2over a image-adic number fieldkwith the Galois groupG. We study the Galois module structure of the ring imageKof integers inK. We determine conditions under which the invariant factors of Kummer orders imageKin imageKof two extensions coincide with each other and give two examples, one of which shows there exist Kummer extensionsKandLwithD(K)=D(L) such that imageKand imageLare not imagepG-isomorphic. The other shows the existence of extensionsFandKsuch that imageFand imageKare isomorphic over imagepGbut not over imagekG.