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The Stacks project

Lemma 9.12.3. Let K/E/F be a tower of algebraic field extensions.

  1. If \alpha \in K is separable over F, then \alpha is separable over E.

  2. if K is separable over F, then K is separable over E.

Proof. We will use Lemma 9.12.1 without further mention. Let P be the minimal polynomial of \alpha over F. Let Q be the minimal polynomial of \alpha over E. Then Q divides P in the polynomial ring E[x], say P = QR. Then P' = Q'R + QR'. Thus if Q' = 0, then Q divides P and P' hence P' = 0 by the lemma. This proves (1). Part (2) follows immediately from (1) and the definitions. \square


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