Let $S$ and $T$ be a noetherian connected normal scheme, say. Let $K$ (resp. $L$) be the function field of $S$ (resp. $T$). Assume that $char(L)=0$. Let $f: S\to T$ be a smooth surjective morphism. Is it true that $f$ factors as $$S\to T'\to T$$ where $S\to T'$ is smooth and surjective with geometrically connected generic fibre and $T'\to T$ is a finite morphism?
(A candidate for $T'$ could be the following scheme: Let $F$ be the algebraic closure of $L$ in $K$. Let $T'$ be the normalization of $T$ in $F$. Then the morphism $f$ factors through $T'$, but I do not know how to prove that the resulting morphism $S\to T'$ is smooth and surjective.)
My main interest is the case where $L$ is a number field and $T$ is a dense open subscheme of $Spec(O_L)$.
