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

TL;DR

This paper explores the non-unique factorization properties of polynomials over residue class rings of integers, revealing infinite elasticity and complex factorization behaviors in these algebraic structures.

Contribution

It characterizes the factorization behavior of polynomials in Z_{p^n}[x], showing the monoids of monic polynomials have infinite elasticity and diverse factorizations.

Findings

Monoids of monic polynomials in Z_{p^n}[x] have infinite elasticity.

Every positive integer m can be realized as a factorization length in these monoids.

Factorization of arbitrary polynomials reduces to monic polynomial factorization.

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.