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Let K be a finite extension of with ramification index e, and let L/K be a finite abelian p-extension with Galois group 0393 and ramification index pn. We give a criterion in terms of the ramification numbers ti for a fractional ideal of the valuation ring S of L not to be free over its associated order In particular, if tn [tn/P] pn-1e then the inverse different can be free over its associated order only when ti = -1 (mod pn) for all i. We give three consequences of this. Firstly, if S) is a Hopf order and S is S)-Galois then ti ~ -1 (mod pn) for all i. Secondly, if K = kr, L = km+r are Lubin-Tate division fields, with m &#x3E; r and k ~ Qp, then S is not free over S). Thirdly, these extensions km+r /kr admit two Hopf Galois structures exhibiting different behaviour at integral level. 1991 Mathematics Subject Classification. 11S23, 11R33, 11S31, 16VV30.

Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications