= Real affine plane scheme points
{title2=$\operatorname{Spec}\mathbb R[x,y]$}
The points of the real <affine plane> as a <scheme> are its <prime ideals>: the zero ideal, the principal ideals of irreducible real polynomials, and its <maximal ideals>. The latter have <residue field> $\mathbb R$ or $\mathbb C$, by the <Zariski lemma> and the <real closed field> property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed <generic points> and closed points absent from the real locus. Its topology is the <Zariski topology>, and its <structure sheaf> has sections $\mathbb R[x,y]_h$ on each <principal open subscheme> $D(h)$.
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