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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The ordinary quaternions over a Pythagorean field
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by Burton Fein and Murray Schacher PDF
Proc. Amer. Math. Soc. 60 (1976), 16-18 Request permission

Abstract:

Let L be a proper finite Galois extension of a field K and let D be a division algebra with center K. If every subfield of D properly containing K contains a K-isomorphic copy of L, it is shown that K must be Pythagorean, $L \cong K(\sqrt { - 1} )$, and D is the ordinary quaternions over K. If one assumes only that every maximal subfield of D contains a K isomorphic copy of L, then, under the assumption that [D : K] is finite, it is shown that K is Pythagorean, $L = K(\sqrt { - 1} )$, and D contains the ordinary quaternions over K.
References
  • I. N. Herstein, On a theorem of A. A. Albert, Scripta Math. 29 (1973), no. 3-4, 391–394. MR 435137
  • I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs, No. 15, Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR 0227205
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 16-18
  • MSC: Primary 12D15; Secondary 12A80, 16A40
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0417139-3
  • MathSciNet review: 0417139