Given $X\subseteq F[x]$ where $F$ is a field, how to prove that there exist a splitting field of $X$ over $F$?
In the case that $X$ is finite, I think the answer can be solved using Kronecker's Theorem, which states that every for every polynomial $f(x)\in F[x]$ there exist an extension field $E$ of $F$ containing a root of $f(x)$. Letting $f(x)$ be the product of all polynomial in $X$, one can proof by induction over $\deg f(x)$ that there is an extension field $K$ of $F$ containing a root of $f(x)$ and, therefore, an extension field $E$ of $F$ such that $f(x)$ splits in linear factors over $E[x]$; from this the result.
But what if $X$ is not finite? In general, how to proceed?