Let $A$ be a normal ring with quotient field $K$. Let $L/K$ be a finite separable extension.
Let $E\_1/K$ and $E\_2/K$ be extensions of $K$ contained in
$L$. Let $B\_1$ (resp. $B\_2$) be the normalization of $A$ in $E\_1$ (resp. $E\_2$) and let $C$ be
the normalization of $A$ in $L$. Then 
$$K\subset E\_i\subset L\ \mbox{and}\ A\subset B\_i\subset C\ \mbox{for}\ i=1, 2.$$
Let $\mathfrak P$ be a prime ideal of $C$. Define $\mathfrak p\_1:=\mathfrak P\cap B\_1$, 
$\mathfrak p\_2:=\mathfrak P\cap B\_2$ and $\mathfrak p:=\mathfrak P\cap A$. Then, on the 
level of residue fields, we have 
$$k(\mathfrak p)\subset k(\mathfrak p\_i)\subset k(\mathfrak P)\ \mbox{for}\ i=1, 2.$$


*Assume that $B\_1$ and $B\_2$ are finite etale $A$-algebras.*


**Questions:**


a) Does $L=E\_1E\_2$ (composite field) imply $k(\mathfrak P)=k(\mathfrak p\_1) k(\mathfrak p\_2)$?


b) Does $K=E\_1\cap E\_2$ imply $k(\mathfrak p)=k(\mathfrak p\_1)\cap k(\mathfrak p\_2)$?


**Comments:**


a) For my application it would be enough to know the answers in the special case where $E\_1/K$ and $E\_2/K$ are Galois extensions.


b) Somebody told me that the answer to a) would be ``no'', if we omitted the hypothesis that the algebras $B\_i/A$ are etale, even in the case where $K$ is a number field.



Composition and intersection of residue fields
a) Consider the canonical map $B\_1\otimes\_A B\_2\to C$. It is finite ($C$ is finite over $A$) and is surjective on the generic fiber. The spectrum of the lefthand side is a disjoint union of normal schemes because of the étale hypothesis. This implies that the spectrum of $C$ is actually one of these connected components. So $\mathrm{Spec} C\to \mathrm{Spec}(B\_1)\times\_A \mathrm{Spec}(B\_2)$ is a closed immersion. This implies easily a). 


b) Counterexample: take two quadratic extensions of $\mathbb Z$ defined by $\sqrt{a}$ and $\sqrt{a+p}$ with $a$ an integer prime to $p$. They have the same residue field at $p$.


[**Add**] In a), Galois hypothesis is not needed. The extensions $\mathbb Z[\sqrt{p}]$ and $\mathbb Z[\sqrt{ap}]$ with $a$ not a square mod $p$, give a non-étale counterexample to a). 


