Given an irreducible polynomial $f\in F[x]$ of degree $n$ and letting $E$ be its splitting field, is it true that every root $\alpha$ of $f$ has degree $n$ so $F(\alpha)=E$? Furthermore $[E:F]=n-s$ where $s$ is the number of repeated roots of $f$?
I've read that $[E:F]$ can be up to $n!$ but don't understand how that can be possible.