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Definition of normal extension: an algebraic extension $K$ of $F$ is normal extension if every irreducible polynomial in $F[x]$ that has one root in $K$ actually splits in $K[x]$.

Let $K$ be a normal extension of the field $F$ of finite degree.Let $E$ be a subfield of $K$ containing $F$. Prove that $E$ is a normal extension of $F$ if and only if every $F$-isomorphism of $E$ onto $K$ is an $F$-automorphism of $E$?

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    I think people here will be generally more willing to help you if you show them that you've sufficiently thought about this question by telling them how you would go about or start the proof. Otherwise, it just looks like a homework problem you're just throwing at us to solve on your behalf.2011-07-30

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Hint for "if": Let $f(x)$ be an irreducible polynomial over $F$, and let $a\in E$ be a root. Let $b$ be conjugate to $a$. Find an automorphism of $K$ that maps $a$ to $b$.

Hint for "only if": If $a\in E$, and $\sigma\colon E\to K$ is an $F$-isomorphism of $E$ into $K$, what can you say about $\sigma(a)$?