I'm wondering definition of "E is a splitting over F"

Question

I know the definition of splitting field $E$ for $f(x)\in F[x]$ but I am confused by this definition: $E$ is a spliiting over $F$ (not $f(x)$).

I want to know that... please answer!

Answer 1

Definition: Let $F$ be a field with algebraic closure $\bar{F}$. Let $$A=\{f_i(x)\mid i\in I\}$$ be a collection of polynomial in $F[x]$. A field $E\leq\bar{F}$ is the splitting field of $A$ over $F$ if $E$ is the smallest subfield of $\bar{F}$ containing $F$ and all the zeroes in $\bar{F}$ of each of the $f_i(x)$ for $i\in I$. A field $K\leq\bar{F}$ is called an splitting field over $F$ if it is the splitting field of some set of polynomials in $F[x]$