Any field that is finitely generated as a ring is a finite field.

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# Small rings

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Any field that is finitely generated as a ring is a finite field.

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This is a version of the “Nullstellensatz over the integers”. Let be the field that is finitely generated as a ring (ie, as a -algebra). If has characteristic , then is finitely generated as an algebra over and we can apply the usual Nullstellensatz. If has characteristic , the universal map is injective and so is torsion-free and hence flat as a -module. Therefore, the map is open, which is a contradiction as its image must only be the generic point of .

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