A Note on Two Dimensional Division Ring Extensions

Part of the Contemporary Mathematicians book series (CM)


A division subring Γ of a division ring Δ is said to be Galois in Δ (and Δ is Galois over Γ) if Γ is the set of invariants (or fixed points) of a group of automorphisms acting in Δ. The two dimensional Galois extensions of a division ring have been determined by Dieudonné.2 In this note we shall show that if Γ is a division ring of characteristic ≠ 2 which is finite dimensional over its center Ψ and Δ contains Γ and has left dimensionality [Δ:Γ] L = 2 then Δ is Galois over Γ On the other hand, we shall construct a class of examples where [Δ: Γ] L = 2 = [Δ:Γ] R , Γ of characteristic ≠ 2 and Δ is not Galois over Γ.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. S. Amitsur, “Non-commutative cyclic fields,” Duke Mathematical Journal, vol. 21 (1954), pp. 87–106.CrossRefGoogle Scholar
  2. [2]
    A. S. Amitsur and J. Levitzki, “Minimal identities for algebras,” Proceedings of the American Mathematical Society, vol. 1 (1950), pp. 449–463.CrossRefGoogle Scholar
  3. [3]
    C. C. Chevalley, The Algebraic Theory of Spinors, New York, 1954.Google Scholar
  4. [4]
    J. Dieudonné, “Les extensions quadratiques des corps non-commutatifs et leurs applications,” Acta Mathematica, vol. 87 (1952), pp. 175–242.CrossRefGoogle Scholar
  5. [5]
    N. Jacobson, The Theory of Rings, New York, 1943.Google Scholar
  6. [6]
    N. Jacobson, “Structure of alternative and Jordan bimodules,” Osaka Mathematics Journal, vol. 6 (1954), pp. 1–71.Google Scholar
  7. [7]
    I. Kaplansky, “Rings with a polynomial identity,” Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 575–580.CrossRefGoogle Scholar
  8. [8]
    I. Kaplansky, “Forms in infinite-dimensional spaces,” Academie Brasiliera de Ciencias, vol. 22 (1950), pp. 1–16.Google Scholar
  9. [9]
    G. Koethe, “Schiefkörper unendlichen Ranges über dem Zentrum,” Mathematische Annalen, vol. 105 (1931), pp. 15–39.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  1. 1.Yale UniversityUSA

Personalised recommendations