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

TL;DR

This paper constructs characteristic q fields with prescribed Brauer dimensions and explores how these dimensions behave under finitely-generated field extensions, revealing infinite Brauer p-dimensions under certain conditions.

Contribution

It demonstrates the existence of fields with specific Brauer dimensions and characterizes possible sequences of Brauer and absolute Brauer p-dimensions in characteristic zero fields.

Findings

Existence of fields with prescribed Brauer dimensions and infinite absolute Brauer p-dimensions.

Finitely-generated transcendental extensions can have infinite Brauer p-dimensions for certain primes.

Any compatible sequence of Brauer and absolute Brauer p-dimensions can be realized in characteristic zero fields.

Abstract

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.