The question asks the following:
Let $f\in F[x]$ be an irreducible polynomial of degree 4, and let $E$ be a splitting field of $f$ over $F$. Can $Gal(E/F)$ be isomorphic to the symmetric group $S_3$.
My guess is that it cannot be, and my approach is to condition on the number of distinct roots. But my difficulty is to argue for the case when the number of distinct roots is 3.
For three distinct roots, I know $f(x) = (x-r_1)^2(x-r_2)(x-r_3)$. I would guess that the key is to show that the Galois group cannot act transitively on these sets of roots, but I cannot figure out exactly how to show that. Any hint on this problem?