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In Mumford's Red Book of Varieties and Schemes, page 102, he gave the example of the closed but not rational points (that is to say points having residue field the complex field and not the real field) of the cubic $y^2=x^3−x$ on the real field : I have some difficulty to recover by elementary methods the figure he traced.

Especially, he seems to imply that these closed points formed the region $y^2>x^3−x$ in the real plane (which looks like the cylinder he pictured in the projective plane). Can somebody give me a simple explanation ? (I suppose the maximal ideals of the spectrum of the algebra defined by the cubic have to be parametrized the right way ?)
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Closed but not rational points of a real cubic

In Mumford's Red Book of Varieties and Schemes, page 102, he gave the example of the closed but not rational points (that is to say points having residue field the complex field and not the real field) of the cubic $y^2=x^3−x$ on the real field : I have some difficulty to recover by elementary methods the figure he traced.

Especially, he seems to imply that these closed points formed the region $y^2>x^3−x$ in the real plane. Can somebody give me a simple explanation ? (I suppose the maximal ideals of the spectrum of the algebra defined by the cubic have to be parametrized the right way ?)