Counting Irreducible Polynomials

Let $N_q(n)$ be the number of irreducible monic polynomials in $\mathbb{F}_q[x]$ of degree $n$. First prove that $$q^n = \sum_{d|n} d\cdot N_q(d).$$ Then, you can use the additive version of the Möbius inversion formula with $H(n)=q^n$ and $h(n)=nN_q(n)$, so that $H(n)=\sum_{d|n} h(d)$ implies that $h(n)=\sum_{d|n}\mu(\frac{n}{d})H(d)$.

You may also have a look at A Classical Introduction to Modern Number theory, by Ireland and Rose, page 84.