Small Galois groups that encode valuations

Ido Efrat, Jan Minac

Published 2011-05-12, updated 2011-12-15Version 2

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.