Assume I am given a polynomial $f$ with coefficients from $\mathbb{Z}_p$ (for prime $p$) of degree $d$.
We know that if $f$ is not the constant zero function, then it has at most $d$ roots.
Question: Is it true that $f$ always has exactly $d$ roots?
If so, then how to show this? If not, then a counter-example would be appreciated.