Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variable prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, monic, deg}(f)=m \right\rbrace|. $$
I want to prove the following statements:
(A) $q$ relatively prime to $m$ $\Longrightarrow$ $q \mid c_m$
(B) $\exists$ a monic polynomial $C_m \in \mathbb{Z}[X]$ with $\text{deg}(C_m)=m$ so that $c_m=\frac{C_m(q)}{m}$ for all $q$.
Hints:
A: We can let the additive group of the field $\Bbb{F}_q$ act on the set of monic irreducible polynomials of degree $m$ as follows. Let $f(x)$, a monic irreducible of degree $m$, and $\alpha\in\Bbb{F}_q$ be arbitrary. Then prove that $f(x+\alpha)$ is also a monic irreducible, and that this defines a group action. The upshot is to prove that if an irreducible monic has a non-trivial stabilizer, then necessarily $\gcd(m,q)\ge p$. Here you can use the fact that all the element of the field have additive order $p$.
B: It looks like you already solved this one.