Lemma 9.15.6. Let $E/F$ be an algebraic extension of fields. If $E$ is generated by $\alpha _ i \in E$, $i \in I$ over $F$ and if for each $i$ the minimal polynomial of $\alpha _ i$ over $F$ splits completely in $E$, then $E/F$ is normal.

Proof. Let $P_ i$ be the minimal polynomial of $\alpha _ i$ over $F$. Let $\alpha _ i = \alpha _{i, 1}, \alpha _{i, 2}, \ldots , \alpha _{i, d_ i}$ be the roots of $P_ i$ over $E$. Given two embeddings $\sigma , \sigma ' : E \to \overline{F}$ over $F$ we see that

$\{ \sigma (\alpha _{i, 1}), \ldots , \sigma (\alpha _{i, d_ i})\} = \{ \sigma '(\alpha _{i, 1}), \ldots , \sigma '(\alpha _{i, d_ i})\}$

because both sides are equal to the set of roots of $P_ i$ in $\overline{F}$. The elements $\alpha _{i, j}$ generate $E$ over $F$ and we find that $\sigma (E) = \sigma '(E)$. Hence $E/F$ is normal by Lemma 9.15.5. $\square$

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