Proposition 6

For any field $K$, the set of monic irreducible polynomials in $K[X]$ is infinite.

Indeed, given an arbitrary nonempty finite family $(p_i)$ ($1 \leq i \leq n$) of distinct monic irreducible polynomials, the polynomial $\left( \prod_{i=1}^n p_i \right) + 1$ is not invertible, and any monic irreducible factor $q$ of this polynomial is necessarily distinct from all the $p_i$, otherwise it would divide 1.