Proposition. equivalent definitions of normal extensions [2205]

Let $K / F$ be an algebraic extension. TFAE:

1. $K / F$ is normal.
2. Any irreducible polynomial in $F[x]$ that has a root in $K$ splits completely in $K$.
3. There exists a subset $S \subseteq K$ such that $K = F(S)$ and $m_{\alpha, F}(x)$ splits completely in $K$ for every $\alpha \in S$.
4. $K$ is the splitting field for a subset of $F[x]$.
5. Fix an algebraic closure of $F$ with $F \subseteq K \subseteq \overline{F}$. (For example, $\overline{F} = \overline{K}$.) Any $F$-embedding $\varphi : K \to \overline{F}$ satisfies $\varphi(K) = K$.
6. Fix an algebraic closure of $F$ with $F \subseteq K \subseteq \overline{F}$. Any $F$-embedding $\varphi : K \to \overline{F}$ satisfies $\varphi(K) \subseteq K$.