Characterising local fields by mostly Galois-theoretic data

Some of the fundamental problems of anabelian geometry are related to characterising fields by Galois-theoretic data. Old results here include axiomatisations of individual fields within certain classes (such as axiomatising global fields by their absolute Galois groups as in the Neukirch-Uchida Theorem) as well as, more importantly for us, the Galois-theoretic characterisation of the so-called p-adically closed fields (a class encompassing the finite extensions of the p-adic numbers) among all fields.

I will discuss how a similar characterisation of local fields fails in positive characteristic as observed by Efrat-Fesenko, and how to obtain a satisfactory positive result by adding some additional information concerning p-torsion in the Brauer group.