Let $f(x) \in F[X]$ a polynomial with degree 5 and Galois Group $S_5$. Show that $f(x)$ is separable and irreducible over $F$.
For irreducibility, I tried by contradiction $$ f(x) = g(x) * h(x) $$ where $g(x), h(x) \in F[X]$ and $g(x)$ is irreduccible. I don't know how to continue, any help?