D is a central division algebra over F. We know that we can always find a maximal subfield K inside D such that K/F is separable. I want to know can we always make it Galois?
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I believe you are just asking if every central division algebra over F is a crossed product. This is not true, and the first example was given in:
Amitsur, S. A. "On central division algebras." Israel J. Math. 12 (1972), 408-420. MR 318216 DOI: 10.1007/BF02764632
A survey of Amitsur's contributions on division algebras which mentions this point in particular is the introduction by Saltman starting on page 109 of:
Amitsur, S. A. Selected papers of S. A. Amitsur with commentary. Part 2. Edited by Avinoam Mann, Amitai Regev, Louis Rowen, David J. Saltman and Lance W. Small. American Mathematical Society, Providence, RI, 2001. xx+615 pp. ISBN: 0-8218-2925-4 MR 1866637 Google
answered Mar 5, 2010 at 22:46
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1$\begingroup$ I agree with this answer of Jack Schmidt. Since I know Pete likes thinking in terms of the Brauer group, I mention the paper of Kiani and Ramezan-Nassab, "Crossed Product Conditions for Central Simple Algebras in Terms of Splitting Fields". The following is proven there (in Pete's notation): Let $n$ be the degree of $D$. $D$ is a crossed product if and only if there exists a Galois extension $L/F$, such that $[D] \in Br(L/F)$, $[L:F] = mn$, $GCD(m,n) = 1$, and $Gal(L/F)$ has a normal subgroup of order $m$. $\endgroup$– MartyMar 6, 2010 at 15:55
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$\begingroup$ Whoops - just noticed that the question was asked by Taisong Jing, and edited by Pete Clark. Hopefully Taisong Jing also appreciates the above characterization. $\endgroup$– MartyMar 6, 2010 at 15:57