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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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Please don't post questions in the imperative; if you would say where you are stuck or why you are having trouble, you will probably have a better shot at a good and useful response. In this case, since "normal" has many different equivalent definitions, you should say what your definition of "normal extension" is (in addition to rephrasing this as a question, rather than an order). –  Arturo Magidin Jul 30 '11 at 5:17
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You know, adding a question mark at the end of "Prove that..." doesn't really turn it into a question, nor does it tell us what you are having trouble with. –  Arturo Magidin Jul 30 '11 at 5:34
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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. –  r.g. Jul 30 '11 at 5:53

1 Answer 1

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 $\sigma(a)$?

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Thanks for your hints,there are really helpful! –  Youli Jul 30 '11 at 20:54

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