[Math] Fundamental theorem of algebra for finite fields

Question

Over the real numbers, a polynomial with degree $n$ has at most $n$ roots, regardless of whether it's a single variate or a multivariate polynomial. Is the same true when the polynomial is over a finite field like $\mathbb{Z}_p$?

Answer 1

Multivariate polynomials over an infinite field can have infinitely many roots, as pointed out by other users.

As for the univariate case, the answer is yes: if $f$ is a (univariate) polynomial over a field $K$ and $a\in K$ is a root, then we can use the division algorithm to show that $f(x)=(x-a)q(x)$ for some polynomial $q$ over $K$.

Note that this isn't the fundamental theorem of algebra, which says that every complex (univariate) polynomial of degree $n$ has exactly $n$ roots, counting multiplicity, in the complex numbers.