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 nonzero 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 m ∈ ℕ, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.

Key Words:

• Elasticity
• Finite ring
• Monoid
• Non-unique factorization
• Polynomial
• Zero-divisor

2000 Mathematics Subject Classification:

• Primary 13M10
• Secondary 13P05, 11T06, 20M14

ACKNOWLEDGMENT

The authors wish to thank F. Halter-Koch for useful comments.

Notes

Communicated by R. Wiegand.