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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 $\mathbb{Z}$.

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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.

André Nicolas
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    Nice example - IIRC there are even polynomials that are reducible over $\mathbb{F}_p$ for all $p$ but not reducible over $\mathbb{Z}$? (Edit: and Wikipedia suggests that $x^4+1$ does the trick: https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields ) – Steven Stadnicki Mar 27 '16 at 06:13