Assume that $(A,m)$ is a Noetherian normal local domain, $K = Quot(A) \subset E, F$ Galois extensions of $K$. If $B=\overline{A}^{E}$, $C=\overline{A}^F$, and $D=\overline{A}^{EF}$ and we choose primes $m_B, m_C$, and $m_D$ (in the corresponding rings) over $m$, then is it true that the separable part of residue field of $m_D$ is generated by the separable parts of the residue fields of $m_B$ and $m_C$ over the residue field of $m$?

The phrase 'separable part' is intended to describe the maximal separable extension of the residue field of $m$ contained in the residue field of $q$ where $q$ is one of $m_B, m_C, $ or $m_D$.

Thanks.