A polynomial with a root in $\mathbb{F}_p \ \forall p$, where $p$ is prime, but no root in $\mathbb{Z}$

Question

Give an example of a polynomial $f(x) \in \mathbb{Z}[x]$ which has a root in every finite field $\mathbb{F}_p$, but no root in $\mathbb{Z}$.

Answer 1

We show that the polynomial $(x^2-2)(x^2-3)(x^2-6)$ works. This clearly has no integer roots. It obviously has roots in the $2$-element field and the $3$-element field. We show it also has a root in $\mathbb{F}_p$ for any prime greater than $3$.

Let $p$ be such a prime. If $2$ is a quadratic residue of $p$, then the equation has a solution in $\mathbb{F}_p$. This is also the case if $3$ is a quadratic residue of $p$. And if neither $2$ nor $3$ is a quadratic residue of $p$, then $6$ is a quadratic residue of $p$, and we are finished.