Let $K\subset L$ be a field extension and let $K\subset F_1,F_2,...,F_n\subset L$ be proper intermediate fields. Consider the "mixed" affine space $\mathbb{A}_{(F_i)}:=\prod_{i=1}^n F_i$ instead of $\mathbb{A}_K^n$ (or $\mathbb{A}_L^n$ for that matter) and consider zero sets of polynomials $f\in K[X_1,X_2,...,X_n]$ in $\mathbb{A}_{(F_i)}$. From the perspective of classic algebraic geometry, it only seems natural to look for such a generalization.

**Has this specific type of problems been studied/discussed in more detail anywhere and, if yes, under what wording?** A reference will be most appreciated. I am mostly interested in the case of algebraic number fields, but other examples are most welcome as well.

Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math., 2010. $\endgroup$ – ACL Jun 25 '13 at 6:07