Polynomials that split and splitting fields of a polynomial or a family of polynomials

Let $E\mathrm{/}F$ be a field extension and let $0\mathrm{\ne }f\in F[x]$ have degree $n$. The polynomial $f$ splits over $E$ if there are $c\in F^{\times }$ and ${\alpha }_1,\dots ,{\alpha }_n\in E$ such that $f(x)=c{\prod }_{j=1}^n(x-{\alpha }_j)$ in $E[x]$, with repetitions allowed. When $n=0$, the product is empty, so every nonzero constant polynomial splits over $F$.

For a family $\mathcal{F}$ of nonzero polynomials in $F[x]$, a splitting field of $\mathcal{F}$ over $F$ is a field extension $E\mathrm{/}F$ such that every member of $\mathcal{F}$ splits over $E$ and $E$ is generated over $F$ by all roots in $E$ of all polynomials in $\mathcal{F}$. A splitting field of the one-element family $\{f\}$ is called a splitting field of $f$. For the empty family, the set of roots is empty and its splitting field is $F$.