Division algebras of prime degree with infinite genus

S. V. Tikhonov

Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 264-267

Résumé

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 $[\mathcal D']\in \mathrm {Br}(F)$, where $\mathcal D'$ is a central division $F$-algebra having the same maximal subfields as $\mathcal D$. For any prime $p$, we construct a division algebra of degree $p$ with infinite genus. Moreover, we show that there exists a field $K$ such that there are infinitely many nonisomorphic central division $K$-algebras of degree $p$ and any two such algebras have the same genus.