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 \(\mathbf{gen} ({\mathcal {D}}\otimes {\mathcal {C}}) \ne \mathbf{gen} ({\mathcal {D}}' \otimes {\mathcal {C}})\).

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Correspondence to Sergey V. Tikhonov.

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Tikhonov, S.V. On genus of division algebras. manuscripta math. 164, 321–325 (2021). https://doi.org/10.1007/s00229-020-01184-4

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Mathematics Subject Classification

  • Primary 16K20
  • Secondary 16K50