Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to do an exercise that asks me to calculate the numer of irreducible polynomials of degree $12 $ over $\mathbb{F}_p$.

The exercise have 3 parts and I have done the first two parts that askd me to find drew the field diagram from $\mathbb{F}_p$ to $\mathbb{F}_{p^12}$ (that is these two fields and their subfields and the connection between those subfields), in part $b$ I showed that the number of primitive elements of the extension is $p^{12}-p^{4}-p^{6}+p^{2}$.

Can someone please help me with the last part of the exercise ? I tried finding the connection between the number I have found on part $b$ and the numer of irreducible polynomials (of degree $12 $ over $\mathbb{F}_p$) and failed.

Also, I know that there's something called "Moebius inversion formula" that can be of help, but I did not study it and since this isn't a general question I would rather not resort to using it.

share|cite|improve this question
Each irreducible polynomial of degree 12 has twelve roots $\alpha$. All of them with the property that $\mathbb{F}_p[\alpha]=\mathbb{F}_{p^{12}}$. So if you know the number of such $\alpha$:s then the number of irreducible polynomials is... – Jyrki Lahtonen Jun 4 '12 at 19:37
@JyrkiLahtonen - at leat the number I found int part b ? – Belgi Jun 4 '12 at 19:39
In a second thought maybe it is the number I found on part b divided by 12 ? Since different irreducable don't share a root and all the roots are different since every finite field is perfect ? – Belgi Jun 4 '12 at 19:43
Correct the second time! – Jyrki Lahtonen Jun 4 '12 at 20:01
This question has been asked several times in the past few weeks, in various forms. Something in the air? – Lubin Jun 4 '12 at 20:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.