Related question.

Let $A$ be a ring, and let $B$ be an $A$-algebra which is projective and finite as an $A$-module. Then the trace $B \rightarrow A$ can be defined. Let's say that $B$ is separable over $A$ if the trace induces an isomorphism of $A$-modules $B \rightarrow \operatorname{Hom}_A(B,A)$. For the purposes of this question, let's say that $B$ is an etale $A$-algebra if it is finite, projective, and separable.

Is it true that every etale $\mathbb Z$-algebra is of the form $\mathbb Z^n$? This is stated in these notes, example 1.1.13(ii).

It does not seem so easy to prove. If this is true, then combined with the much easier result, "if $L/K$ is an unramified extension of number fields, then $\mathcal O_L$ is an etale $\mathcal O_K$-algebra," we would obtain an algebraic proof that there are no nontrivial unramified extensions of $\mathbb Q$. I had thought that the only known proofs of this result relied on Minkowski's bound for the discriminant. Perhaps any proof that etale $\mathbb Z$-algebras are of the form $\mathbb Z^n$ must rely on this fact.

finite étale $A$-algebra, because there are also étale $A$-algebras that are not finite. $\endgroup$ – R. van Dobben de Bruyn Jul 23 '18 at 19:591more comment