[Math] The galois group of a polynomial of degree 3 is either $A_3$ or $S_3$

Question

Hungerford -Algebra p.271

Let $E/F$ be a Galois extension where $E$ is a splitting field for a separable irreducuble polynomial $f$ over $F$ whose roots are $a_1,a_2,a_3$.

Let $\Phi:Gal(E/F)\rightarrow S_n$ be the natural group monomorphism permuting roots of $f$.

Then the order of '$Gal(E/F)$ is divisible by $3$, henve it is a normal subgroup of $S_3$.

These are all I know. How do I prove that the Galois group is eithet S3 or A3?

Answer 1

Hint: Why must $\text{Gal}(E/F)$ act transitively on the roots of $f$? What are the transitive subgroups of $S_3$?

Please let me know if you would like further elaboration. Hope this helps!