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?
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:
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