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?