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Let $k$ be an algebraic closure of $\mathbb{F}_p$. Let $X$ be a connected smooth quasi-projective $k$-scheme. If $K$ is an algebraically closed field containing $k$, is the prime-to-$p$ etale fundamental group of $X$ isomorphic to that of its base change to $K$?

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Yes, the canonical map $X_K\rightarrow X$ induces an isomorphism on $\pi_1^{(p)}$. You can find here a detailed proof of a slightly more general statement.

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    $\begingroup$ for future use, in case the link gets broken, this is : Aaron Landesman, Invariance of the fundamental group under base change between algebraically closed fields. $\endgroup$ – Niels Apr 23 '19 at 8:14
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    $\begingroup$ A friend contacted me asking for a permanent home for this write up, so it is now posted at arxiv.org/abs/2005.09690 Also see mathoverflow.net/questions/257722/… $\endgroup$ – Aaron Landesman May 21 at 1:43

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