Small rings

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

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Small rings

One thought on “Small rings

  1. nachodarago says:

    This is a version of the “Nullstellensatz over the integers”. Let A be the field that is finitely generated as a ring (ie, as a \mathbb{Z}-algebra). If A has characteristic p>0, then A is finitely generated as an algebra over \mathbb{F}_p and we can apply the usual Nullstellensatz. If A has characteristic 0, the universal map \mathbb{Z}\to A is injective and so A is torsion-free and hence flat as a \mathbb{Z}-module. Therefore, the map \mathrm{Spec}(A)\to\mathrm{Spec}(\mathbb{Z}) is open, which is a contradiction as its image must only be the generic point of \mathrm{Spec}(\mathbb{Z}).

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