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Let $X$ be an inverse limit of integral, normal affine curves $X_i = Spec(A_i)$ where all the transition maps are finite etale covering maps (so in particular $X$ exists in the category of schemes), and all the curves are over some field $k$. Let $F$ be the function field of $X$, so $F$ is the union of the function fields $F_i$ of each $X_i$, or alternatively the fraction field of $A$, the union of the $A_i$.

Now let $f \in A^* \setminus (A^*)^p$. Let $Y$ be the normalization of $X$ in $F(f^{1/p})$.

Is the morphism $Y \rightarrow X$ constructed above a finite etale cover?

I have some reasons for suspecting the answer might be no, but am very unsure. The problem seems to be that $A$ is not noetherian, I think, and so the normalization might fail to be finite.

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