Let $f(x)$ be a nonconstant polynomial over $\mathbb Z$.


Must $f$ have a zero in $\mathbb F\_p$ for some prime number $p$?


More generally, let $f\_1,\dots,f\_k$ be such polynomials, must there exist a $p$ such that each $f\_i$ has a zero in $\mathbb F\_p$?



Integers mod $p$ "together" algebraically closed
Easy answer to the first question: for $n \gg 0$, we have $f(n) > 1$ or $f(n) < -1$. Hence $f(n)$ is divisible by some prime number $p$, so $n$ is a root mod $p$.


