What is meant by the statement that $L$ is a splitting field for $f \in K[X] ?$

Show that if $f \in K[X]$, then there exists a splitting field for $f$ over $K$. Explain the sense in which a splitting field for $f$ over $K$ is unique.

Determine the degree $[L: K]$ of a splitting field $L$ of the polynomial $f=X^{4}-4 X^{2}+2$ over $K$ in the cases (i) $K=\mathbb{Q}$, (ii) $K=\mathbb{F}_{5}$, and (iii) $K=\mathbb{F}_{7}$.