Let $f(x)$ be irreducible over a field $k$ characteristic zero, with splitting field $E$, and $\alpha$ and $\beta$ be the roots of $f$ in $E$. If the galois group of $E/k$ is abelian prove that $k(\alpha)=k(\beta)$.
Now since the group is abelian the extensions $k(\alpha)$ and $k(\beta)$ are galois and also both have the same degree over $k$. But have can we conclude the statement? Thanks for any help.
