Proving degree $n$ have at last $n$ roots in $F_q[X]$

Question

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?

Answer 1

A field (finite or not) is always an integral domain, hence assuming that $\xi$ is a root of $p(x)=a_n x^n+\ldots+a_0$, then $p(x)=(x-\xi)q(x)$, where the degree of $q(x)$ is just the degree of $p(x)$ minus one. By induction, the number of the roots cannot exceed the degree.