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The Stacks project

Lemma 9.12.12. Let E/k and F/E be separable algebraic extensions of fields. Then F/k is a separable extension of fields.

Proof. Choose \alpha \in F. Then \alpha is separable algebraic over E. Let P = x^ d + \sum _{i < d} a_ i x^ i be the minimal polynomial of \alpha over E. Each a_ i is separable algebraic over k. Consider the tower of fields

k \subset k(a_0) \subset k(a_0, a_1) \subset \ldots \subset k(a_0, \ldots , a_{d - 1}) \subset k(a_0, \ldots , a_{d - 1}, \alpha )

Because a_ i is separable algebraic over k it is separable algebraic over k(a_0, \ldots , a_{i - 1}) by Lemma 9.12.3. Finally, \alpha is separable algebraic over k(a_0, \ldots , a_{d - 1}) because it is a root of P which is irreducible (as it is irreducible over the possibly bigger field E) and separable (as it is separable over E). Thus k(a_0, \ldots , a_{d - 1}, \alpha ) is separable over k by Lemma 9.12.10 and we conclude that \alpha is separable over k as desired. \square


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