In his Algebraic surfaces book, Beauville gives a result allowing one to "absorb ramification" for certain maps (see below). There are also something similar one can do with number fields. I would really like to know how I should think about these results, (that is, are they specific to these two situations or are they vastly more general.) I'm really hoping for a reference (to EGA?) where such a version is stated, assuming it is true.

This question naturally lends itself to questions 2,3; which are hopefully covered in the same place as #1 (assuming again that they are true).

$\textbf{1.}\textrm{ Absorbing ramification}$

If $K|k$ is a finite extension of number fields, then there are infinitely many finite extensions $E|k$, so that $K\cap E = k$ and $EK|E$ is unramified. (You can get such extensions from the approximation lemma.) (Does this fact have a name?)

$\textbf{Setup:}$ Let $X \xrightarrow{\phi} Y$ be a finite, faithfully flat map of noetherian schemes. Feel free to assume conditions on $X, Y, \phi$, as you need.

Can we have something similar for faithfully flat maps $\phi$? More specifically, let $\phi, X,Y$ be as in the setup. Does there exist $Z \xrightarrow{\tau} Y$ a finite, faithfully flat map so that the base change along $\tau$ is etale. Is there a lemma (like the approximation lemma) showing that there are "many" such maps in good conditions, and allowing us to control the ramification of $\tau$.

-In certain cases, we can absorb ramification for a fibration of a smooth surface (Beauville, p73 Lemma VI.7):

$\textbf{2.}\textrm{ Maximal etale subextensions}$

Let $\phi, X,Y$ be as in the setup.

Does there exist a maximal etale subextension? That is, can we factor $\phi$ as $X \rightarrow Y_{et} \xrightarrow{\psi} Y$ with $\psi$ etale, and so that $\psi$ is maximal amongst all such factorizations. That is, if $\phi$ also factors as $$X \rightarrow Y' \xrightarrow{\psi'} Y$$ with $\psi'$ etale, then the map $Y_{et} \xrightarrow{\psi} Y$ factors as $Y_{et} \rightarrow Y' \xrightarrow{\psi'} Y$.

$\textbf{3.}\textrm{ Galois closure}$

The natural question is take $X, Y, \phi$ be as our initial setup, (assume that $\phi$ is separable, $X,Y$ are projective over a field, integral) and ask if there a galois closure $Z \rightarrow X$, so that $Z/Aut(Z/Y) \cong Y$ naturally. However, it's not even clear to me that the automorphism group would be finite (or more generally that the quotient exists as a scheme) - or there exists such $Z$ which is "minimal".

These questions bear a strong relationship to theorems true for number fields. One issue of course is that we use composition of number fields in number theory and in algebraic geometry we have base change along maps which are a little different operations.

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