I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$.
One of strategies to get it is finding an irreducible polynomial $f(X) \in \mathbb{Q}[X]$ which has exactly three real roots. But, I can't do it.
Can you give me examples of quintic polynomial $f(X) \in \mathbb{Q}[X]$ which has exactly three real roots and is irreducible over $\mathbb{Q}$ $??$