On genus of division algebras


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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  1. 1.

    Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: Division algebras with the same maximal subfields. Russ. Math. Surv. 70(1), 83–112 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: The finiteness of the genus of a finite-dimensional division algebra, and some generalizations. Isr. J. Math. (to appear). arXiv:1802.00299

  3. 3.

    Pierce, R.S.: Associative Algebras. Graduate Texts in Mathematics, vol. 88. Springer, New York (1982)

    Book  Google Scholar 

  4. 4.

    Saltman, D.J.: Lectures on Division Algebras. Amer. Math. Soc, Providence (1999)

    Book  Google Scholar 

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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