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.



Algebraic varieties in "mixed" affine spaces
The product of Weil restrictions $P = \prod\_{i=1}^n {\rm{R}}\_{F\_i/K}(\mathbf{A}^1\_{F\_i})$ is naturally a closed subscheme of the direct product $\prod\_{i=1}^n {\rm{R}}\_{L/K}(\mathbf{A}^1\_L) = {\rm{R}}\_{L/K}(\mathbf{A}^n\_L)$. For any closed subscheme $Z \subset \mathbf{A}^n\_K$, we also get a closed subscheme ${\rm{R}}\_{L/K}(Z\_L) \subset {\rm{R}}\_{L/K}(\mathbf{A}^n\_L)$ and can see where it meets $P$. Those intersections are the $K$-schemes you're wondering about (e.g., the $K$-points are precisely the loci inside $\prod F\_i$ defined by the vanishing of several polynomials in $n$ variables over $K$). 


From the viewpoint of Weil restriction of affine schemes of finite type, this looks like a rather special kind of construction. It isn't clear why it should be useful or a natural thing to consider from the viewpoint of classical algebraic geometry (though Weil restriction is very useful). Is there a situation you have in mind for which this is interesting to consider?


