In algebraic number theory, one chooses for each finite étale $\mathbb{Q}$-algebra $K$ a finite $\mathbb{Z}$-algebra $\mathcal{O}_K$. Usually one simply speaks of the finite $\mathbb{Q}$-algebras which are fields, but the étale $\mathbb{Q}$-algebras are just products of these.

Now $- \otimes \mathbb{Q}$ takes the ring of integers $\mathcal{O}_K$ to $K$. Moreover, any integral basis for $\mathcal{O}_K$ is a $\mathbb{Q}$-basis for $K$. In terms of descent theory, we might think of $\mathcal{O}_K$ as a $\mathbb{Z}$-form of $K$. Indeed, this seems similar to the situation for Galois descent, or descent in general. However, as far as I know, descent theory applies to Galois extensions $L/K$ of fields. In that case, a descent datum for an $L$-algebra $A$ is the structure of an $G$-module, where $G$ is the Galois group of the extension, but here, any map of $K$ fixing $\mathcal{O}_K$ seems to also fix $K$.

I guess I'll still ask: is it possible to do descent theory in a way that applies to this context? More precisely, what kind of descent data would we need to recover $\mathcal{O}_K$ from $K$?