= Solution
A finite extension $L/K$ is <normal field extension>[normal] when every <irreducible polynomial> in $K[T]$ having one root in $L$ splits completely over $L$.
Suppose first that $L/K$ is normal. Choose $\alpha_1,\ldots,\alpha_m$ with $L=K(\alpha_1,\ldots,\alpha_m)$, and let $f_j$ be the <minimal polynomial> of $\alpha_j$ over $K$. Every $f_j$ splits in $L$ by normality, and $L$ is generated by their roots because it is already generated by the particular roots $\alpha_j$. Thus $L$ is the <splitting field> of $f_1\cdots f_m$.
Conversely, suppose that $L$ is the splitting field of $f\in K[T]$. Every $K$-embedding $\sigma:L\to\overline K$ permutes the roots of $f$, so $\sigma(L)=L$. By the embedding characterization of <normal field extension>[normality], $L/K$ is normal. This proves the <finite normal extension as a splitting field> criterion.
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