More fun in 2015, Part 8

Algebra Level 5

How many polynomials \(p(x)\) with integer coefficients (PICs) divide \(x^{2015}-1\), in the sense that \(x^{2015}-1=p(x)q(x)\) for some PIC?

Here is some background info for those who have not studied this kind of Number Theory yet: For any positive integer \(n\), we define the cyclotomic polynomial \(\Phi_n(x)=\prod(x-w)\), where the product is taken over all primitive nth roots of unity, \(w\). We are told that \(\Phi_n(x)\) is a PIC and that it cannot be factored into two PICs of positive degree (\(\Phi_n(x)\) is "irreducible").

Extra Credit question: Explain why the cyclotomic polynomials have integer coefficients.

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