When I try to find the number of irreducible polynomials (of degree n) over a finite field I first look for the number of $\alpha \in \mathbb{F}_{q^n}$ such that $\mathbb{F}_{q^n}=\mathbb{F}_{q}[\alpha]$. And then i just divide the number of $\alpha$'s by n.
Is this always right? What happens when n divides the characteristic of the field? If $\alpha$ verifies $\mathbb{F}_{q^n}=\mathbb{F}_{q}[\alpha]$ can $\alpha$ be a double root of a polynomial?