More fun in 2015, Part 8

Algebra Level 5

How many polynomials p(x)p(x) with integer coefficients (PICs) divide x20151x^{2015}-1, in the sense that x20151=p(x)q(x)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 nn, we define the cyclotomic polynomial Φn(x)=(xw)\Phi_n(x)=\prod(x-w), where the product is taken over all primitive nthn^\text{th} roots of unity, ww. We are told that Φn(x)\Phi_n(x) is a PIC and that it cannot be factored into two PICs of positive degree (Φn(x)\big(\Phi_n(x) is "irreducible").\big).


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

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