= Complexification fibres of a real scheme
{title2=$\operatorname{Spec}(\kappa(p)[s]/(s^2+1))$}
For a <scheme> over $\mathbb R$, its <base change of a morphism of schemes> to $\mathbb C$ has <scheme-theoretic fibre> over a point $p$ equal to $\operatorname{Spec}(\kappa(p)\otimes_{\mathbb R}\mathbb C)$. This is the spectrum of $\kappa(p)[s]/(s^2+1)$. It has two points if $-1$ is a square in the <residue field>, and otherwise one point with a quadratic <field extension>. In both cases it is reduced, because the quadratic polynomial has no repeated root in <characteristic zero>. The distinction applies to <generic points> as well as <closed points>.
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