Consider the following:
Let $E/K$ be a separable field extension of degree $p$ ($p$ a prime). Suppose $f\in K[x]$ is an irreducbile polynomial which has more than one root in $E$. Show $f$ splits in $E[x]$.
I've tried a couple different ideas, but I haven't been able to make anything out of it. Here is what I know for certain, though: Any of the roots in $E$ must be a primitive element for $E$. Since $E/K$ is separable, we conclude $f$ must be separable and therefore its splitting field $F$ (containing $E$) is a Galois extension of $K$.
If I could show $Aut(F/E)$ is normal in $Gal(F/K)$ I'd be done, for then $E/K$ is normal. I know $Aut(F/E)$ has index $p$ in $Gal(F/K)$. This doesn't seem to get me anywhere though since I don't know much about $Gal(F/K)$.
I am having a difficult time finding a way to use the "has more than one root in E" hypothesis.
Can anybody give me a nudge in the right direction?