TITLE: Prove $x^5 + x^4 + x^3 + x^2 + 1 = 0$ has no rational solution
QUESTION [5 upvotes]: Prove $x^5 + x^4 + x^3 + x^2 + 1 = 0$ has no rational solution
I want to prove it by Proof by Contradiction, but I am not sure how to proceed with the proof.

REPLY [4 votes]: I think that Jorge Fernández' solution is the most effective. 
So, here we go for an overkill. Since $p(x)$ splits as:
$$ (x^2+x-1)(x^3-x-1)$$
over $\mathbb{F}_3$, it has no rational root.