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Let p be a prime.

(a) Determine the number of irreducible polynomials over Z_{p}of the form x^{2}+ ax + b.

(b) Determine the number of irreducible quadratic polynomials over Z_{p}.

The attempt at a solution

A nonzero, nonunit polynomial f(x) in Z_{p}[x] is irreducible if it equals the product of two polynomials in Z_{p}[x], one of them being a unit of Z_{p}[x]. But does Z_{p}[x] have any units? I find it hard to imagine that there are polynomials g(x) and h(x) in Z_{p}[x] with g(x)h(x) = 1, unless of course g(x) = h(x) = ±1.

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# Homework Help: Counting Number of Irreducibles

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