Let $F$ be a field and $f\in F[x]$ a polynomial that is irreducible, separable and of degree $4$.
How do I prove that there are not more than $5$ different possibilities for the Galois group Gal($f$) (not counting isomorphisms)?
I found multiple answers/sites that determine exactly what the Galois groups can be. But those were a little too advanced for me, since I just got introduced to Galois theory. I think that's also the reason why the question says 'not more than 5' instead of being more specific.
So maybe somebody can help?
Edit: @DietrichBurde has helped a lot here below by giving a source that says that Gal($f$) must be a transitive subgroup of $S_4$, of which there are only five. However, it's not clear for me why it must be a transitive subgroup of $S_4$.