Lemma 57.6.3. Let $K$ be a field and set $X = \mathop{\mathrm{Spec}}(K)$. Let $\overline{K}$ be an algebraic closure and denote $\overline{x} : \mathop{\mathrm{Spec}}(\overline{K}) \to X$ the corresponding geometric point. Let $K^{sep} \subset \overline{K}$ be the separable algebraic closure.

The functor of Lemma 57.2.2 induces an equivalence

\[ \textit{FÉt}_ X \longrightarrow \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets}. \]compatible with $F_{\overline{x}}$ and the functor $\textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets} \to \textit{Sets}$.

This induces a canonical isomorphism

\[ \text{Gal}(K^{sep}/K) \longrightarrow \pi _1(X, \overline{x}) \]of profinite topological groups.

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