A Note on Two Dimensional Division Ring Extensions
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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.
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