Quintic polynomial problem without analysis

Question

Just wondering: Is there a purely algebraic proof of the existence of an irreducible quintic polynomial over $\mathbb{Q}$ with exactly three real roots? Sure, it's easy to give concrete examples, but I don't see how the proof of the polynomial having exactly three real roots can avoid using completeness of $\mathbb{R}$ in the form of some intermediate/mean value theorem-type methods (that's what I mean by purely algebraic).