Counting Irreducible Polynomials

Question

I'm investigating irreducible polynomials over finite fields at the moment, and I wanted to know if there is a formula for the number of irreducible polynomials of degree n over a fixed finite field $\mathbb{F}_q$. Wolfram MathWorld gives the formula \begin{equation} \frac{1}{n}\sum_{d|n}\mu(\frac{n}{d})q^d \end{equation} However, neither it nor the OEIS page it links to offers any proof for this as far as I can tell, and I'm kind of confused by the presence of a divisor sum; I don't see why such a sum would appear in dealing with these polynomials.

So how does one prove this formula?

This question has been answered previously on this site. See for example here — Dilip Sarwate, Dec 19 '11 at 15:53
Anyhow, this looks like the Mobius inversion formula.
If $f(n)$ is the number of irreducible polynomials, this formula suggests that $q^n= \sum_{d|n} df(d) $, which doesn't look obvious to me... — N. S., Dec 19 '11 at 15:56
@N.S.: That's obvious once you know that $x^{q^n}-x$ is the product of all the (monic) irreducible polynomials over $\mathbb{F}_q$ of degree dividing $n$. — Chris Eagle, Dec 19 '11 at 16:05
@Dilip: Sorry; for some reason I didn't notice. Thanks for pointing that out. — Will Dana, Dec 19 '11 at 17:13
Note: The Möbius function is defined by $$\mu(n) = \begin{cases} 0 & \text{when } n \text{ has one or more repeated prime factors} \ 1 & \text{when } n=1 \ (-1)^k & \text{when } n \text{ has exactly } k \text{ distinct prime factors} \end{cases} $$ — hardmath, Nov 20 '17 at 01:24

score 12 · Accepted Answer · edited Dec 20 '11 at 13:02

12

Let $N_q(n)$ be the number of irreducible monic polynomials in $\mathbb{F}_q[x]$ of degree $n$. First prove that $$q^n = \sum_{d|n} d\cdot N_q(d).$$ Then, you can use the additive version of the Möbius inversion formula with $H(n)=q^n$ and $h(n)=nN_q(n)$, so that $H(n)=\sum_{d|n} h(d)$ implies that $h(n)=\sum_{d|n}\mu(\frac{n}{d})H(d)$.

edited Dec 20 '11 at 13:02

Srivatsan

26,311

answered Dec 19 '11 at 16:04

Álvaro Lozano-Robledo

15,390

1

I always liked to show my students this, whenever I taught algebra. – Lubin Dec 20 '11 at 03:56
It is indeed a very neat application of Mobius inversion! – Álvaro Lozano-Robledo Dec 20 '11 at 04:06
Thank you for pointing this out. – Will Dana Dec 20 '11 at 06:32

score 2 · Answer 2 · edited Aug 25 '22 at 19:13

2

You may also have a look at A Classical Introduction to Modern Number theory, by Ireland and Rosen, page 84.

edited Aug 25 '22 at 19:13

DanLewis3264

2,321

answered Dec 19 '11 at 21:29

Ehsan M. Kermani

9,228

Counting Irreducible Polynomials

2 Answers2

Linked