On the divisors of

We examine two natural questions concerning the polynomial divisors of x n − 1 : “For a given integer n, how large can the coefficients of divisors of x n − 1 be?” and “How often does x n − 1 have a divisor of every degree between 1 and n?” We consider the latter question when x n − 1 is factored in both Z [ x ] and F p [ 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 F p [ x ] .