The valuation criterion for normal basis generators

If L/K is a finite Galois extension of local fields, then we say that the valuation criterion VC(L/K) holds if there is an integer d such that every element x is an element of L with valuation d generates a normal basis for L/K. Answering a question of Byott and Elder, we first prove that VC(L/K) holds if and only if the tamely ramified part of the extension L/K is trivial and every non-zero K[G]-submodule of L contains a unit of the valuation ring of L. Moreover, the integer d can take one value modulo [L:K] only, namely-d(L/K)-1, where d(L/K) is the valuation of the different of L/K. When K has positive characteristic, we thus recover a recent result of Elder and Thomas, proving that VC(L/K) is valid for all extensions L/K in this context. When char K=0, we identify all abelian extensions L/K for which VC(L/K) is true, using algebraic arguments. These extensions are determined by the behaviour of their cyclic Kummer subextensions.