On the divisors of xn−1

We examine two natural questions concerning the polynomial divisors of xn−1: “For a given integer n, how large can the coefficients of divisors of xn−1 be?” and “How often does xn−1 have a divisor of every degree between 1 and n?” We consider the latter question when xn−1 is factored in both Z[x] and Fp[x]. The primary tools used in our investigation arise the study of the anatomy of integers. We also make use of several results on the size of the multiplicative order function (which stem from Hooleyʼs conditional proof of Artinʼs Primitive Root Conjecture) in our work over Fp[x].