Small Galois groups that encode valuations
  Let $p$ be a prime number and let $F$ be a field containing a root of unity of order $p$. We prove that a certain very small canonical Galois group $(G_F)_{[3]}$ over $F$ encodes the valuations on $F$ whose value group is not $p$-divisible and which satisfy a variant of Hensel's lemma. 
Ido Efrat and Jan Minac
math.NT