  We prove that every non-trivial valuation on an infinite superrosy field of
positive characteristic has divisible value group and algebraically closed
residue field. In fact, we prove the following more general result. Let $K$ be
a field such that for every finite extension $L$ of $K$ and for every natural
number $n>0$ the index $[L^*:(L^*)^n]$ is finite and, if $char(K)=p>0$ and $f:
L \to L$ is given by $f(x)=x^p-x$, the index $[L^+:f[L]]$ is also finite. Then
either there is a non-trivial definable valuation on $K$, or every non-trivial
valuation on $K$ has divisible value group and, if $char(K)>0$, it has
algebraically closed residue field. In the zero characteristic case, we get
some partial results of this kind.
  We also notice that minimal fields have the property that every non-trivial
valuation has divisible value group and algebraically closed residue field.
