Geodesic
On the size of the genus of a division algebra
Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 69-99

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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. 
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     author = {Vladimir I. Chernousov and Andrei S. Rapinchuk and Igor A. Rapinchuk},
     title = {On the size of the genus of a division algebra},
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Vladimir I. Chernousov; Andrei S. Rapinchuk; Igor A. Rapinchuk. On the size of the genus of a division algebra. Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 69-99. http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a4/