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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.

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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

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