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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 Albert, Scripta Math. 29 (1972), 391-394. MR 0435137 (55:8098)
  • [2] -, Noncommutative rings, Carus Math. Monographs, vol. 15, Math. Assoc. Amer., distributed by Wiley, New York, 1968. MR 37 #2790. MR 0227205 (37:2790)

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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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