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The ordinary quaternions over a Pythagorean field


Authors: Burton Fein and Murray Schacher
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
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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 [Enhancements On Off] (What's this?)

  • [1] I. N. Herstein, On a theorem of A. A. Albert, Scripta Math. 29 (1973), no. 3–4, 391–394. Collection of articles dedicated to the memory of Abraham Adrian Albert. MR 0435137
  • [2] I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR 0227205

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0417139-3
Keywords: Division algebra, Pythagorean field
Article copyright: © Copyright 1976 American Mathematical Society

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