I want to know how hany monic irreducible polynomials of degree $3$ there are in a field $\mathbb{F}_q$.
The whole number of monic polynomials of degree three is $q^3$. Now I want to find out how many reducible polynomials there are among them. We can split any reducible $f$ in one and only one of the following ways:
$$ f \ = \ (X-\alpha)(X^2+bX+c) \qquad \text{or} \qquad f \ = \ (X-\alpha)(X-\beta)(X-\gamma) $$
- In the first case, we take the number of monic irreducible polynomials of degree $2$ and multiply them by $q$, which gives us $\frac12(q^3-q^2)$.
- To count the polynomials of the other kind, I thought it would be helpful to split up this set as $$ \{(X-\alpha)^3 \ : \ \alpha \in \mathbb{F}_q \} \quad \sqcup \quad \{ (X-\alpha)^2(X-\beta) \ : \ \alpha \neq \beta \ \} \quad \sqcup \quad \{ (X-\alpha)(X-\beta)(X- \gamma) \ : \ \alpha, \beta, \gamma \text{ all differ }\} $$ This would give us the following number of reducible polynomials of this shape: $$ q + \frac12q(q-1)+\frac16q(q-1)(q-2) $$
Would you think that I will get the right answer if I'd calculate the value below? $$ q^3 - \frac12(q^3-q^2) - \left( q + \frac12q(q-1)+\frac16q(q-1)(q-2) \right) $$