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Irreducible polynomial problems

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Determine the number of monic irreducible polynomials of degree 4 in F_q without using the Moebius Inversion Formula.

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This solution helps with questions involving irreducible polynomials.

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Let us first calculate the number of reducible monic polynomials of degree 4 over F_q. Each of them has one of the following forms
〖(x-a)〗^4, for some a from F_q,
〖(x-a)〗^3(x-b), for distinct a and b from F_q ,
〖(x-a)〗^2 〖(x-b)〗^2 , for distinct a and b from F_q,
〖(x-a)〗^2(x-b)(x-c), for distinct a,b,c from F_q,
〖(x-a)〗^2f(x), where f(x) is an irreducible monic polynomial over F_q of the degree 2,
(x-a)f(x), ...

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