Separable extension..

**ynj** · September 2nd 2009, 09:35 PM

Let $F$ be a field with characteristic 0. $E$ is a finite field extension of $F$ . Prove that $E$ is a separable extension...
I know that for an $\alpha$ , if the minimal polynomial $f(x)$ splits on $E$ , then $f(x)$ is separable on $E$ . But why $f(x)$ splits?

**ThePerfectHacker** · September 3rd 2009, 11:57 AM

Originally Posted by ynj

Let $F$ be a field with characteristic 0. $E$ is a finite field extension of $F$ . Prove that $E$ is a separable extension...
I know that for an $\alpha$ , if the minimal polynomial $f(x)$ splits on $E$ , then $f(x)$ is separable on $E$ . But why $f(x)$ splits?

You are misunderstanding what "seperable" means. An irreducible polynomial $f(x)\in F[x]$ is "seperable" over $F$ iff $f(x)$ has no repeated roots in its splitting field. Now $\alpha \in E$ is seperable iff the minimal polynomial for $\alpha$ is seperable over $F$ . This does not mean that the miniminal polynomial must split over $E$ .

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