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This article is cited in 12 scientific papers (total in 12 papers)
On the size of the genus of a division algebra
Vladimir I. Chernousova, Andrei S. Rapinchukb, Igor A. Rapinchukc a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
b Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
c Department of Mathematics, Harvard University, Cambridge, MA, 02138 USA
Abstract:
Let $D$ be a central division algebra of degree $n$ over a field $K$. One defines the genus $\mathbf {gen}(D)$ as the set of classes $[D']\in \mathrm {Br}(K)$ in the Brauer group of $K$ represented by central division algebras $D'$ of degree $n$ over $K$ having the same maximal subfields as $D$. We prove that if the field $K$ is finitely generated and $n$ is prime to its characteristic, then $\mathbf {gen}(D)$ is finite, and give explicit estimations of its size in certain situations.
Received: September 7, 2015
Citation:
Vladimir I. Chernousov, Andrei S. Rapinchuk, Igor A. Rapinchuk, “On the size of the genus of a division algebra”, Algebra, geometry, and number theory, Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 292, MAIK Nauka/Interperiodica, Moscow, 2016, 69–99; Proc. Steklov Inst. Math., 292 (2016), 63–93
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https://www.mathnet.ru/eng/tm3691https://doi.org/10.1134/S0371968516010052 https://www.mathnet.ru/eng/tm/v292/p69
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