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 $K$?



Number Rings and (Galois) Descent
The categorical Galois theory of [Borceux and Janelidze](https://ncatlab.org/nlab/show/Galois+Theories) given in chapter 4 for commutative rings applies to your situation.


In particular, it applies to any 'effective Galois descent morphism' defined as follows:



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> **Definition 1.** Let $\mathcal{C}$ be a category. An arrow $f:X\to Y\in \mathsf{Hom}\_\mathcal{C}$ is an *effective descent morphism* iff the pullback along $f$ functor $-\_f:\mathcal{C}/Y\to\mathcal{C}/X$ is monadic.
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> **Definition 2.** Let $\sigma:R\to S$ be a morphism of rings. Write $\eta$ for the unit of the adjunction $$\mathsf{Sp}\_S:(\mathsf{S}\text{-Alg})^{op}\rightleftarrows\mathsf{Prof}/\mathsf{Sp}(S):\mathcal{C}\_S,$$ where $\mathsf{Sp}\_S$ and $\mathcal{C}\_S$ are defined by $$\mathsf{Sp}\_S(A)=\big(\mathsf{Sp}(A)\longrightarrow\mathsf{Sp}(S)\big),$$ $$\mathcal{C}\_S(X,f)=\mathsf{Hom}\big((X,f),(\coprod\_MS/M,p)\big),$$ with $p:\coprod\_MS/M\to\mathsf{Sp}(S)$ the projection of the Pierce structural space of the ring $S$. An $R$-algebra $A$ is *split* by $\sigma$ when the morphism $$\eta\_{S\otimes\_RA}:\mathcal{C}\_S\mathsf{Sp}\_S(S\otimes\_R A)\longrightarrow S\otimes\_RA$$ is an isomorphism.
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> **Definition 3.** A morphism of rings $\sigma:R\to S$ is of **effective Galois descent** iff
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> 1. $\sigma$ is an effective descent morphism in $\mathsf{Ring}^{op}$, and
> 2. For every object $(X,\psi)\in\mathsf{Prof}/\mathsf{Sp}(S)$, the $R$-algebra $\mathcal{C}\_S(X,\psi)$ is split by $\sigma$.
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They then give the following Galois theorem for commutative rings.



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> **Theorem 1.** Let $\sigma:R\to S$ be an effective Galois descent morphism, with $Gal[\sigma]$ the corresponding Galois groupoid in the category of profinite spaces. Then there exists an equivalence of categories $$\big(\mathsf{Split}\_R(\sigma)\big)^{op}\simeq\mathsf{Prof}^{Gal[\sigma]}$$ between the dual of the category of $R$-algebras split by $\sigma$ and the category of internal covariant presheaves on $Gal[\sigma]$ in the category of profinite topological spaces.
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The reference to étale stuff was for étale morphisms of topological spaces, but your situation should fall under the scope of above theorem.


