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Is there any convenient way to understand the étale fundamental group of projective spaces over finite fields, in particular, the étale fundamental group of $\mathbf{P}^2_{\mathbf{F}_q}$?

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    $\begingroup$ This is just the Galois group $\operatorname{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q) $. $\endgroup$
    – abx
    Feb 17, 2021 at 10:38
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    $\begingroup$ The fundamental group of $\mathbb{P}^n_{\overline{\mathbb{F}_q}}$ can be computed by noting that the fundamental group $\mathbb{P}^1_{\overline{\mathbb{F}_q}}$ is trivial, that the fundamental group of a direct product is the direct product of the fundamental groups and that fundamental group is a birational invariant. $\endgroup$
    – Faris
    Feb 17, 2021 at 14:09
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    $\begingroup$ See for example Tag 0BTX combined with the computation of $\pi_1(\mathbf P^n_{\bar k})$. (See here for a fun Riemann–Hurwitz-free proof when $n=1$.) $\endgroup$ Feb 17, 2021 at 18:03
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    $\begingroup$ The fundamental group does not respect products. That requires either completeness or coprime characteristic, so it's ok here, but it's not true in general. If it were true, then the fundamental group of a group scheme would be abelian. But it's not for $\mathbb A^1$. @Faris $\endgroup$ Feb 17, 2021 at 22:48

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