On Brauer p-dimensions and absolute Brauer p-dimensions of Henselian fields

Abstract

This paper determines the Brauer p-dimension Brdp(K) and the absolute Brauer p-dimension abrdp(K) of a Henselian valued field (K,v), for a prime p≠char(Kˆ), under restrictions on the residue field Kˆ, such as the condition abrdp(Kˆ)=0. It describes the set Σ0 of sequences abrdp(E),Brdp(E), p∈P, where P is the set of prime numbers and E runs across the class of Henselian fields with char(Eˆ)=0 and a projective absolute Galois group GEˆ. Specifically, Σ0 contains a sequence ap,bp∈N∪{0,∞}, p∈P, whenever a2≤2b2 and ap≥bp, for each p. Similar results are obtained in characteristic q>0.