Suppose $F$ is a finite splitting field over $K$ of $X=\lbrace f_i(x)\rbrace_{i\in I}$, some infinite set. Is there necessarily a finite set $Y\subseteq X$ such that $F$ is a finite splitting field of $Y$?
I'm curious if there is a way to generalize: $F$ is a splitting field over $K$ of a finite set $\lbrace f_1,…,f_n\rbrace$ of polynomials in $K[x]$ if and only if $F$ is a splitting field over $K$ of the single polynomial $f=f_1f_2⋯f_n$.
The example I have in mind is $\mathbb{C}$ over $\mathbb{R}$. Clearly $\mathbb{C}$ is the splitting field for $X=\lbrace ax^2+bx+c\mid a,b,c\in \mathbb{R}, b^2-4ac<0 \rbrace$ and but also for just $x^2+1$.
It is a classic result in Galois theory that a finite extension $F/K$ is the splitting field of a single polynomial (i.e. generated by all the roots of a single polynomial) if and only if $F/K$ is normal, i.e. every irreducible polynomial in $K$ with a root in $F$ splits completely. See this post, for instance.