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The Stacks project

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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