Number of irreducible monic polynomials of degree $3$ over $\Bbb F_p$, $p$ prime

Question

Good morning :)

So I had an exercise, which asks "how many monic irreducible polynomials are there of degree 3 over the finite polynomial ring $\Bbb F_p[X]$, where $p$ is some prime?"

I was wondering if you can do this via counting?

I know that there are $ p(p-1)/2$ monic irreducible polynomials of degree 2 over $\Bbb F_p[X]$.

And then I counted how many reducible polynomials of degree 3 there are, which is:

A reducible polynomial of degree 3 has to have at least one linear factor.

Then we have 4 options:

1) $(X-a)(X^2 + bX + c)$, where the quadratic term is irreducible. => There are $p^2(p-1)/2$ polynomials of this form.

2) $(X-a)^3$ => p polynomial of this form.

3) $(X-a)^2*(X-b)$, where $a$, $b$ distinct => $p(p-1)/2$. polynomials of this form.

4) $(X-a)(X-b)(X-c)$, where $a, b, c$ pairwise distinct => $p(p-1)(p-2)/6$ polynomials of this form.

And in total I have $p^3$ possible monic polynomials of degree 3 with coefficients in $\Bbb F_p$.

So then the number of irreducible, monic polynomials of degree $3$ over $\Bbb F_p$ is

$p^3 - (p^2(p-1)/2 + p + p(p-1)/2 + p(p-1)(p-2)/6) = p(p-1)(p+3)/3$ when I count it.

However I know the correct solution is supposed to be $(p^3 - p)/3$. Now I do not see where I made the mistake, if anybody here can spot my mistake I would be very thankful.

Thank you in advance. :)

Answer 1

You say

3) $(X−a)^2(X−b)$, where $a, b$ distinct => $p(p−1)/2$. polynomials of this form.

The division by $2$ there suggests that order doesn't matter, but it does. Namely, $(X - a)^2(X-b) \neq (X - b)^2(X-a)$.

The correct number there is $p(p-1)$. You will find that this fixes your problem: https://www.wolframalpha.com/input/?i=p%5E3%E2%88%92(p%5E2(p%E2%88%921)%2F2+%2B+p+%2B+p(p%E2%88%921)+%2B+p(p%E2%88%921)(p%E2%88%922)%2F6)