On the behaviour of Brauer pp-dimensions under finitely-generated field extensions

The present paper shows that if $q \in \mathbb P$ or $q = 0$, where $\mathbb P$ is the set of prime numbers, then there exist characteristic $q$ fields $E _{q,k}\colon \ k \in \mathbb N$, of Brauer dimension Brd$(E _{q,k}) = k$ and infinite absolute Brauer $p$-dimensions abrd$_{p}(E _{q,k})$, for all $p \in \mathbb P$ not dividing $q ^{2} - q$. This ensures that Brd$_{p}(F _{q,k}) = \infty$, $p \dagger q ^{2} - q$, for every finitely-generated transcendental extension $F _{q,k}/E _{q,k}$. We also prove that each sequence $a _{p}, b _{p}$, $p \in \mathbb P$, satisfying the conditions $a _{2} = b _{2}$ and $0 \le b _{p} \le a _{p} \le \infty$, equals the sequence abrd$_{p}(E), {\rm Brd}_{p}(E)$, $p \in \mathbb P$, for a field $E$ of characteristic zero.Comment: LaTeX, 14 pages: published in Journal of Algebra {\bf 428} (2015), 190-204; the abstract in the Metadata updated to fit the one of the pape

On the residue fields of Henselian valued stable fields

AbstractLet (K,v) be a Henselian valued field satisfying the following conditions, for a given prime number p: (i) central division K-algebras of (finite) p-primary dimensions have Schur indices equal to their exponents; (ii) the value group v(K) properly includes its subgroup pv(K). The paper shows that if Kˆ is the residue field of (K,v) and Rˆ is an intermediate field of the maximal p-extension Kˆ(p)/Kˆ, then the natural homomorphism Br(Kˆ)→Br(Rˆ) of Brauer groups maps surjectively the p-component Br(Kˆ)p on Br(Rˆ)p. It proves that Br(Kˆ)p is divisible, if p>2 or Kˆ is a nonreal field, and that Br(Kˆ)2 is of order 2 when Kˆ is formally real. We also obtain that Rˆ embeds as a Kˆ-subalgebra in a central division Kˆ-algebra Δˆ if and only if the degree [Rˆ:Kˆ] divides the index of Δˆ