On genus of division algebras

Abstract

The genus $\mathbf{gen}({\mathcal {D}})$ of a finite-dimensional central division algebra ${\mathcal {D}}$ over a field F is defined as the collection of classes $[{\mathcal {D}}']\in \text {Br}(F)$, where ${\mathcal {D}}'$ is a central division F-algebra having the same maximal subfields as ${\mathcal {D}}$. We show that the fact that quaternion division algebras ${\mathcal {D}}$ and ${\mathcal {D}}'$ have the same maximal subfields does not imply that the matrix algebras $M_l({\mathcal {D}})$ and $M_l({\mathcal {D}}')$ have the same maximal subfields for $l>1$. Moreover, for any odd $n>1$, we construct a field L such that there are two quaternion division L-algebras ${\mathcal {D}}$ and ${\mathcal {D}}'$ and a central division L-algebra ${\mathcal {C}}$ of degree and exponent n such that $\mathbf{gen} ({\mathcal {D}}) = \mathbf{gen} ({\mathcal {D}}')$ but $\mathbf{gen} ({\mathcal {D}}\otimes {\mathcal {C}}) \ne \mathbf{gen} ({\mathcal {D}}' \otimes {\mathcal {C}})$.