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

Christopher Frei, Sophie Frisch

    Research output: Contribution to journalArticlepeer-review

    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.
    Original languageEnglish
    Pages (from-to)1482-1490
    Number of pages9
    JournalCommunications in Algebra
    Volume39
    Issue number4
    DOIs
    Publication statusPublished - 2011

    Keywords

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

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