When the group of automorphisms of an extension of fields acts transitively Let $F$ be a field, $f(x)$ a non-constant polynomial, $E$ the splitting field of $f$ over $F$, $G=\mathrm{Aut}_F\;E$. How can I prove that $G$ acts transitively on the roots of $f$ if and only if $f$ is irreducible?
(if we suppose that $f$ doesn't have linear factor and has degree at least 2, then we can take 2 different roots that are not in $F$, say $\alpha,\beta$, then the automorphism that switches these 2 roots and fixes the others and fixes also $F$, is in $G$, but I'm not using the irreducibility, so where is my mistake?

I think what you needed as a hypothesis was that the extension is Galois. By this, I mean that the polynomial $f(x)$ splits into distinct linear factors in an extension, so I am just assuming the negation of the problem about having repeated roots that other folks were suggesting. In this case, the transitive action on the roots of $f(x)$ should give irreduciblity, as the action permutes the roots of any irreducible factor of $f(x)$.
