Non-unique factorization of polynomials over residue class rings of the integers

Abstract

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to irreducible polynomials in Z_p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every positive integer m, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.