There are infinitely many irreducible polynomials in $\Bbb{F}_p[X]$
Assume that there are finitely many (monic) irreducible polynomials in $\mathbf F_{p}[X]$,$p_{1}(x), p_{2}(x), \dots, p_{n}(x)$. Consider $f(x)=p_{1}(x)p_{2}(x) \cdots p_{n}(x)+1$. Now $f(x)$ is not divisible by any irreducible, and hence is irreducible and is not in the list of irreducibles. Contradiction.