Let $K$ be a field, $t$ a transcendetal element over $K$ and $F'|K(t)$ an infinite Galois extension. Hence I have a tower of extension $K\subset K(t) \subset F' $. Does exist a subextension $F$ of $F'$ such that:
-$F'$ is a finite algebraic extension of $F(t)$?
In general if I have a curve $U$ over a field $K$ with function field $K(U)$ let $F'$ be the maximal galois unramified extension given by the composite of all finite separable extension $L$ of $K(U)$ such that the corresponding cover of curves $C_L\rightarrow U$ is etale. Does exist a subfield $F$ of $F'$ such that:
-$F'$ is a finite algebraic extension of $F(t)$
-$F$ is algebraically closed in $F'$?