Fix an algebraic integer $\alpha$ of degree $n$ such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields. (We can assume $K$ is Galois with non-simple Galois group.) This $\alpha$, by its powers, gives rise to a co-ordinatisation for the affine space $K=\mathbf{Q}^n$. Now what is known about subset of algebraic numbers inside $K$ of a fixed degree $m$ less than $n$? Is this a $\mathbf{Q}$-Zariski closed subset of $K$? Or should we consider subsets of elements of degree less than or equal to $m$?

If we fix the Galois group $G$ and the divisor $m$ of $[K:\mathbf Q]$ is the ideal for degree m elements independent of $K$?