Here is a really nice identity.

Let $\displaystyle F = \text{GF}(q)$ where $\displaystyle q$ is a prime power.
Let $\displaystyle M_n$ be the set of of all monic polynomials of degree $\displaystyle n$.
Let $\displaystyle d[f(x)]$ be the number of monic divisors $\displaystyle f(x)\in F[x]$ has.

Then we have the following identity,
$\displaystyle \sum_{f(x)\in M_n} d[f(x)] = (n+1)q^n$