Lemma 9.15.2 (09HN)—The Stacks project

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Part 1: Preliminaries
Chapter 9: Fields
Section 9.15: Normal extensions
Lemma 9.15.2 (cite)

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Lemma 9.15.2. Let $K/E/F$ be a tower of algebraic field extensions. If $K$ is normal over $F$, then $K$ is normal over $E$.

Proof. Let $\alpha \in K$. 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 $P = QR$. Hence, if $P$ splits completely over $K$, then so does $Q$. $\square$

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