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.



Maximal separable extensions of residue fields
Counterexample based on Hagen's remark: take $A=\mathbb{R}[[t]]$, $B=\mathbb{R}[[\sqrt{t}]]$, $C=\mathbb{R}[[\sqrt{-t}]]$. Both have residue field $\mathbb{R}$, but $D$ contains a square root of $-1$ (namely $\sqrt{-t}/\sqrt{t}$), so its residue field must be $\mathbb{C}$.


