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Math Help - determine irreducible quadratic polynomials

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    determine irreducible quadratic polynomials

    Let p be a prime. Determine the number of irreducible quadratic polynomials over Z_p

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    Quote Originally Posted by mandy123 View Post
    Let p be a prime. Determine the number of irreducible quadratic polynomials over Z_p
    The polynomial x^{p^2} - x factors into a product of monic irreducible polynomials of order dividing 2.
    There are p linear monic polynomials. Let n be the number of monic irreducible quadradic polynomials.
    Then by counting degrees of polynomials in x^{p^2} - x = \prod_{\deg p(x) | 2}p(x)
    We see that p + 2N = p^2 \implies N = \tfrac{1}{2}(p^2 - p).

    (This formula can be generalized to polynomials of degree m by applying Mobius inversion formula)
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    So then what if p is not a prime, what if we are told to
    Determine the number of irreducible quadratic polynomials over Z_p?
    how would that change the answer?
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    Quote Originally Posted by mandy123 View Post
    So then what if p is not a prime, what if we are told to
    Determine the number of irreducible quadratic polynomials over Z_p?
    how would that change the answer?
    Finite fields have orders of power of primes. Thus, your question would be to count the number of irreducible quadradics over \mathbb{F}_q where q=p^n (a power of prime). In this case the same theorem applies i.e. x^{q^2} - x factors into monic irreducible polynomials having order dividing 2.
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