Curves Which Do Not Become Semi-stable after Any Solvable Extension

We show that there is a field F complete with respect to a discrete valuation whose residue field is perfect and there is a finite Galois extension K|F such that there is no solvable Galois extension L|F such that the extension KL|K is unramified, where KL is the composite of K and L. As an application we deduce that that there is a field F as above and there is a smooth, projective, geometrically irreducible curve over F which does not acquire semi-stable reduction over any solvable extension of F .