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Cyclic algebras, complete fields, and crossed products

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Abstract

Letk be a field andn a positive integer. We construct a field extensionK ofk and a cyclic division algebraD of indexn with centerK.

Theorem 1

Let q=char(k). Let M be a subfield of D which is Galois over K of degree m with Galois group H.

  1. 1)

    If q/m then H has a normal q-Sylow subgroup.

  2. 2)

    Iq q ✗ m then H is an abelian group with one or two generators, an extension of a cyclic group by a cyclic group of order e where k contains a primitive e-th root of unity.

Letk(X) be the generic division ring overk of indexn as defined by Amitsur.

Theorem 2

If n is divisible by the square of a prime p≠char(k) and k does not contain a primitive p-th root of unity, then k(X) is not a crossed product.

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Authors and Affiliations

  1. The Technion-Israel Institute of Technology, Haifa, Israel

    Lawrence J. Risman

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  1. Lawrence J. Risman
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Dedicated to the memory of Richard Brauer

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Risman, L.J. Cyclic algebras, complete fields, and crossed products. Israel J. Math. 28, 113–128 (1977). https://doi.org/10.1007/BF02759787

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  • DOI: https://doi.org/10.1007/BF02759787

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