Counting Irreducible Polynomials

I'm investigating irreducible polynomials over finite fields at the moment, and I wanted to know if there is a formula for the number of irreducible polynomials of degree n over a fixed finite field $\mathbb{F}_q$. Wolfram MathWorld gives the formula \begin{equation} \frac{1}{n}\sum_{d|n}\mu(\frac{n}{d})q^d \end{equation} However, neither it nor the OEIS page it links to offers any proof for this as far as I can tell, and I'm kind of confused by the presence of a divisor sum; I don't see why such a sum would appear in dealing with these polynomials.

So how does one prove this formula?

Solution 1:

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 $h(n)=\sum_{d|n}\mu(\frac{n}{d})H(d)$.

Solution 2:

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