On Some Special Factorizations of (1-xn)/(1-x)

Abstract

Let A: a1 < a2 < … < ak be a set of non-negative integers We call the corresponding polynomial the characteristic polynomial, or briefly, the c-polynomial of A. Any polynomial of such a form we call a c-polynomial and any factorization of a c-polynomial into others of the kind we call a c-factorization. If a c-polynomial cannot be factored in this way we call it c-irreducible. In this note we will determine all c-factorizations of the polynomial 1 + x + x2 + … + xn-1, and will find under what circumstances the c-irreducible factors are also irreducible in the usual sense, i.e., irreducible over the field of rationals.