# Small rings

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

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## 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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