# Separable extension..

• Sep 2nd 2009, 10:35 PM
ynj
Separable extension..
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?
• Sep 3rd 2009, 12:57 PM
ThePerfectHacker
Quote:

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 $E$.