Definition 57.55.1. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point lying over the point $s$ of $S$. Let $\kappa (s) \subset \kappa (s)^{sep} \subset \kappa (\overline{s})$ denote the separable algebraic closure of $\kappa (s)$ in the algebraically closed field $\kappa (\overline{s})$.

In this situation the

*absolute Galois group*of $\kappa (s)$ is $\text{Gal}(\kappa (s)^{sep}/\kappa (s))$. It is sometimes denoted $\text{Gal}_{\kappa (s)}$.The geometric point $\overline{s}$ is called

*algebraic*if $\kappa (s) \subset \kappa (\overline{s})$ is an algebraic closure of $\kappa (s)$.

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