Theorem

Let $p$ be prime. If $N_{d}$ denotes the number of monic irreducible polynomials of degree $d$ (where $d\le n$) in $\mathbb{F}_{p}[x]$ then $$p^{n}=\sum_{d|n}dN_{d}$$ which by the Möbius Inversion Formula gives us $$\begin{align*} nN_{n}&=\sum_{d|n}\mu(d)p^{\frac{n}{d}}\\ &\iff\\ N_{n}&=\frac{1}{n}\sum_{d|n}\mu(d)p^{\frac{n}{d}} \end{align*}$$where $\mu$ is the Möbius Function.

Corollary

$N_{n}\ge 1$ for every value of $n\ge1$. i.e. For every value of $n$, there is an irreducible polynomial of degree $n$ in $\mathbb{F}_{p}[x]$.

Intuition

This gives us an analytic formula $$N_{n}=\frac{1}{n}\sum_{d|n}\mu(d)p^{\frac{n}{d}}$$giving us a way to find the number of monic irreducible polynomials of degree $n$ in $\mathbb{F}_{p}[x]$.