The Stacks project

Remark 72.10.6. Let $k$ be a finite field. Let $K/k$ be a geometrically irreducible field extension. Then $K$ is the limit of geometrically irreducible finite type $k$-algebras $A$. Given $A$ the estimates of Lang and Weil [LW], show that for $n \gg 0$ there exists an $k$-algebra homomorphism $A \to k'$ with $k'/k$ of degree $n$. Analyzing the argument given in the proof of Lemma 72.10.5 we see that if $X$ is a quasi-separated algebraic space over $k$ and $X_ K$ is a scheme, then $X$ is a scheme. If we ever need this result we will precisely formulate it and prove it here.