Quintic polynomial with three real roots

Question

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 $\mathbb{Q}$ $??$

Answer 1

You can take $p(x)=x^5-4x-2$. It is irreducible in $\mathbb Q[x]$, by Eisenstein's criterion. And it is easy to deduce from the fact that $p'(x)=5x^4-4$ and from the intermediate value theorem that it has $3$ and only $3$ real roots.