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

Extract

Let A: a₁ < a₂ < … < a_k 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 + … + x^n-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.