Relative brauer group and pro-p galois group of pre-p-henselian fields

Abstract

Let p be a prime number and (F, v) a valued field. In this paper, we find a presentation for the p-torsion part of the Brauer group Br(F), by means of the valuation v. We only assume that F has a primitive pth root of the unity and the residue class field has characteristic not equal to p. This result naturally leads to consider valued fields that we call pre-p-henselian fields. It concerns valuations compatible with Rp, the p-radical of the field. To be precise, Rp is the radical of the skew-symmetric pairing which associates to a pair (a, b) the class of the symbol algebra (F; a, b) in BrF. In our main result, we state that pre-p-henselian fields are precisely the fields for which the Galois group of the maximal Galois p-extension admits a particular decomposition as a free pro-p product.