Integral Galois module structure of some Lubin-Tate extensions

Abstract

Let K be a finite extension of ℚp, and suppose that K/ℚp is ramified and that the residue field of K has cardinality at least 3. Let K(2) be the second division field of K with respect to a Lubin-Tate formal group, and let Γ̃ = Gal(K(2)/K). We determine the associated order in KΓ̃ of the valuation ring script D sign(2) of K(2), and show that script D sign(2) is not free over this order. The integral Galois module structure of certain intermediate fields E of K(2)/K is also considered. In particular, if p ≠ 2 and K has residue field of cardinality p or p2, we show that the valuation ring of E is free over its associated order if and only if E/K is either tamely ramified or a p-extension. We also prove that the valuation ring of any weakly ramified abelian extension of K is free over its associated order. © 1999 Academic Press.