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I want to prove that fields are (étale) $K(\pi,1)$.

One possibility would be to apply "If $Y \to X$ is a connected pro-étale Galois cover such that $Y$ is $K(\pi,1)$, then so is $X$." to $\mathrm{Spec}{K^{sep}} \to \mathrm{Spec}{K}$.

Are there other proofs?

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  • $\begingroup$ Have you looked in Artin-Mazur's Étale homotopy theory book? $\endgroup$
    – K.J. Moi
    Commented Jun 18, 2013 at 8:43
  • $\begingroup$ Yes, and I didn't find anything about this. $\endgroup$
    – user19475
    Commented Jun 18, 2013 at 9:09

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