Here is a really nice identity.

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

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