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Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$.

How to compute $\pi_1^{et}(X)$?

Note. I am only interested in the part not coming trivially from the finite etale covers. For instance, if $S$ is the spectrum of the field of rational numbers, $\pi_1^{et}(X)$ surjects onto the Galois group of $\mathbb{Q}$.

If $S$ is the spectrum of an algebraically closed field of characteristic zero, this group can be computed over the field of complex numbers, and we get $\widehat{\mathbb{Z}}$ as the connected finite etale covers of $X$ are in this case $x\mapsto x^n$ (with $n>0$).

What if $S$ is a dense open of Spec $\mathbb Z$? What if $S$ is of finite type over Spec $\mathbb Z$? What if $S$ is a variety over $\mathbb F_p$?

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    $\begingroup$ What do you mean by 'the part not coming trivially from the finite étale covers'? At any rate, understanding $\pi_1^{\operatorname{\acute et}}(X)$ should be at least as difficult as understanding $\pi_1^{\operatorname{\acute et}}(S)$, so in general it's pretty hard to compute. $\endgroup$ – R. van Dobben de Bruyn Jul 17 '17 at 16:22

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