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I tried searching for this on stack exchange but couldn't get it. I want to compute the number of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_q$

I know one method which uses mobius inversion formula, but I have seen my professor doing something using inclusion-exclusion for particular values of $n$ and $d$. I don't know what he did, but if anyone has any idea about it please let me know or if it is written somewhere then please provide me the reference for the same.

What I remember is he assumed some root, then said it has q^n possibilites and constructed some field, then went on to use inclusion exclusion principle and concluded but I couldn't understand what he said.

Qiaochu Yuan
  • 419,620
permutation_matrix
  • 1,356
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    I believe you ask about this proof: https://www.maa.org/sites/default/files/Chebolu11739.pdf – Mark Aug 28 '22 at 20:41
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    Möbius inversion is "essentially" the inclusion-exclusion. Particular when $n$ is square-free. Several "duplicate targets" or other closely related questions are listed or linked to here. That (and some other threads) specify $q$ to be a prime, but that is irrelevant. – Jyrki Lahtonen Aug 28 '22 at 20:46
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    Here is a very simple example of exclusion-inclusion when $n=6$. – Jyrki Lahtonen Aug 28 '22 at 20:50
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    This shows the Möbius inversion (other threads may be closer to your liking though). – Jyrki Lahtonen Aug 28 '22 at 20:54

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