On the size of the genus of a division algebra

TL;DR

This paper investigates the set of division algebras sharing the same maximal subfields over finitely generated fields, proving finiteness of this set and providing explicit size estimates under certain conditions.

Contribution

It establishes the finiteness of the genus of a division algebra over finitely generated fields and offers explicit bounds on its size in specific cases.

Findings

gen(D) is finite over finitely generated fields

Explicit size estimates for gen(D) in certain situations

Finiteness holds when the degree is prime to the characteristic

Abstract

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K represented by central division algebras D' of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.