# On genus of division algebras

## TL;DR

This paper investigates the properties of the genus of division algebras, showing that sharing maximal subfields does not imply similar properties for matrix algebras or after tensoring with certain algebras, revealing nuanced distinctions.

## Contribution

It demonstrates that having the same maximal subfields does not guarantee similar maximal subfields in matrix algebras or after tensoring, providing new insights into the structure of division algebras.

## Key findings

- Quaternion division algebras with the same maximal subfields can differ in their matrix algebra extensions.
- Existence of fields where tensoring with certain division algebras changes the genus relationship.
- The genus concept does not always extend predictably under matrix or tensor operations.

## Abstract

The genus $gen(D)$ of a finite-dimensional central division algebra $D$ over a field $F$ is defined as the collection of classes $[D']\in Br(F)$, where $D'$ is a central division $F$-algebra having the same maximal subfields as $D$. We show that the fact that quaternion division algebras $D$ and $D'$ have the same maximal subfields does not imply that the matrix algebras $M_l(D)$ and $M_l(D')$ 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 $D$ and $D'$ and a central division $L$-algebra $C$ of degree and exponent $n$ such that $gen(D) = gen(D')$ but $gen(D \otimes C) \ne gen(D' \otimes C)$.