# Maximal subfield inside a central division algebra

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 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 $m$. –  Marty Mar 6 '10 at 15:55