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?



Maximal subfield inside a central division algebra
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](http://www.ams.org/mathscinet-getitem?mr=318216)
[DOI: 10.1007/BF02764632](http://dx.doi.org/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](http://www.ams.org/mathscinet-getitem?mr=1866637)
[Google](http://books.google.com/books?id=p2uvFqAEhLIC&pg=PA109)


