Factoring polynomials in the ring of formal power series over Z

TL;DR

This paper investigates the factorization of polynomials with integer coefficients within the ring of formal power series over Z, providing conditions for reducibility and an explicit factorization algorithm, especially for polynomials with prime power constant terms.

Contribution

It offers new sufficient conditions for reducibility and introduces an explicit factorization algorithm in Z[[x]] for polynomials, including those with prime power constants.

Findings

Established criteria for polynomial reducibility in Z[[x]]

Developed an explicit factorization algorithm

Connected factorization properties to p-adic integers

Abstract

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over Z[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.