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

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

2009-07-03

arxiv.org/abs/0907.0657v2

Communications in Algebra, 39: 1482-1490, 2011

Mathematics

Number Theory

11 pages

Scientific paper

10.1080/00927872.2010.549158

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.

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