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Division algebras with common subfields

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Abstract

We study the partial ordering on isomorphism classes of central simple algebras over a given field F, defined by setting \(A_1 \le A_2\) if \(\deg A_1 = \deg A_2\) and every étale subalgebra of \(A_1\) is isomorphic to a subalgebra of \(A_2\), and generalizations of this notion to algebras with involution. In particular, we show that this partial ordering is invariant under passing to the completion of the base field with respect to a discrete valuation, and we explore how this partial ordering relates to the exponents of algebras.

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Notes

  1. Note that this is somewhat different from the definition given in [14, Section 6]; however it is equivalent in the case of discrete valuations by [14, Lemma 6.2], and the fact that there are no tame and totally ramified division algebras over a complete discretely valued field (see, for example [31, Remark 3.2(a)]. We often assume that the degree of D is prime to the characteristic of F.

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Authors and Affiliations

  1. Bar-Ilan University, Ramat-Gan, Israel

    Eliyahu Matzri & Louis Rowen

  2. Rutgers University, New Brunswick, New Jersey, US

    Daniel Krashen

  3. University of Virginia, Charlottesville, Virginia, US

    Andrei Rapinchuk

  4. The Center for Communications Research, Princeton, New Jersey, US

    David Saltman

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  1. Daniel Krashen
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  2. Eliyahu Matzri
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Correspondence to Louis Rowen.

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This work was supported by the U.S.-Israel Binational Science Foundation (Grant No. 201049) and by the Israel Science Foundation (Grant No. 630/17)

The authors have no conflicts of interest to declare that are relevant to the content of this article. There is no extra data.

The authors thank the referee for correcting some slips and suggesting improved proofs.

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Krashen, D., Matzri, E., Rapinchuk, A. et al. Division algebras with common subfields. manuscripta math. 169, 209–249 (2022). https://doi.org/10.1007/s00229-021-01315-5

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