  Let $F$ be a field complete with respect to a discrete valuation whose
residue field is perfect of characteristic $p>0$. We prove that every smooth,
projective, geometrically irreducible curve of genus one defined over $F$ with
a non-zero divisor of degree a power of $p$ has a solvable point over $F$. We
also 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 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 $F$.

Number Theory;Algebraic Geometry