The valuation criterion for normal basis generators (Q2903274)

Let \(L/K\) be a finite Galois extension of local fields with Galois group \(G\), and let \(v\) denote the discrete valuation on \(L\). Starting with a result of \textit{N.~P. Byott} and \textit{G.~G. Elder} [Bull. Lond. Math. Soc. 39, No. 5, 705--708 (2007; Zbl 1128.11055)] one wants to characterize all such extensions which have the following property, called (VC): There exists some \(d \in \mathbb Z\) such that every element \(x \in L\) with \(v(x)=d\) is a normal generator, i.~e. the conjugates of \(x\) are a \(K\)-basis for \(L\).NEWLINENEWLINEIf (VC) holds, it is shown that \(L/K\) must be totally ramified and that for \(d\) there is only one possible value modulo \([L:K]\). From this the authors obtain the following criterion: (VC) holds for a non-trivial extension \(L/K\) iff the residue characteristic \(p\) is positive, \(L/K\) is totally ramified of degree some power of \(p\), and every non-zero \(K[G]\)-submodule of \(L\) contains some \(u\) with \(v(u)=0\).NEWLINENEWLINEAssuming that \(K\) is of mixed characteristic \((0,p)\), the authors investigate the latter condition for totally ramified Kummer extensions, and from this derive Theorem 1.4, which gives an explicit characterization of (VC) for abelian extensions of local fields.