On the size of the genus of a division algebra

Vladimir I. Chernousov ; Andrei S. Rapinchuk ; Igor A. Rapinchuk

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 69-99

Résumé

Let $D$ be a central division algebra of degree $n$ over a field $K$. One defines the genus $\mathbf {gen}(D)$ as the set of classes $[D']\in \mathrm {Br}(K)$ in the Brauer group 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 $\mathbf {gen}(D)$ is finite, and give explicit estimations of its size in certain situations.