# Quintic polynomial with three real roots

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 $$\mathbb{Q}$$ $$??$$

• A lot of textbooks on Galois theory have them. – Lord Shark the Unknown Apr 15 at 6:50

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.