Abstract :
LetKbe a finite extension of imagep, and suppose thatK/imagepis ramified and that the residue field ofKhas cardinality at least 3. LetK(2)be the second division field ofKwith respect to a Lubin–Tate formal group, and letΓ=Gal(K(2)/K). We determine the associated order inKΓof the valuation ring image(2)ofK(2), and show that image(2)is not free over this order. The integral Galois module structure of certain intermediate fieldsEofK(2)/Kis also considered. In particular, ifp≠2 andKhas residue field of cardinalityporp2, we show that the valuation ring ofEis free over its associated order if and only ifE/Kis either tamely ramified or ap-extension. We also prove that the valuation ring of any weakly ramified abelian extension ofKis free over its associated order.
