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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