My question arises from this.
I'm wondering that if $K$ is a field, $f \in K[X]$ is a separable and not necessarily irreducible polynomial and $E$ is the splitting of $f$ over $K$, then $Gal(E/K)$ acts transitively on the roots of any irreducible factor of $f$.
I think it is possible to make a similar argument to the given in the other question. If $g$ is an irreducible factor, and $\alpha,\beta$ are roots of $g$, we can still make the isomorphism $\sigma:K(\alpha)/K \to K(\beta)/K$ with $\sigma(\alpha)=\beta$, as $g = Irr(\alpha,K) = Irr(\beta,K)$. Then, as $E/K$ is still a normal extension containing $K(\alpha)$ and $K(\beta)$, $\sigma$ can be extended to an automorphism $\overline{\sigma}:E/K \to E/K$, so that $\overline{\sigma} \in Gal(E/K)$ and $\overline{\sigma}(\alpha)=\beta$. Is this right?
