  We show that each local field $\mathbb{F}_q((t))$ of characteristic $p > 0$
is characterised up to isomorphism within the class of all fields of imperfect
exponent at most $1$ by (certain small quotients of) its absolute Galois group
together with natural axioms concerning the $p$-torsion of its Brauer group.
This complements previous work by Efrat-Fesenko, who analysed fields whose
absolute Galois group is isomorphic to that of a local field of characteristic
$p$.
