On Hensel's roots and a factorization formula in \(\mathbb Z[[x]]\) (Q2926272)

The authors investigate the arithmetic properties of \(\mathbb{Z}[[x]],\) the ring of formal power series with integer coefficients. The article completes the previous paper [Int. J. Number Theory 8, No. 7, 1763--1776 (2012; Zbl 1255.13016)] where the authors studied the factorization problem over \(\mathbb{Z}[[x]]\) and established a connection between reducibility and the existence of \(p\)-adic root. In the paper under the review, the authors give an explicit factorization over \(\mathbb{Z}[[x]]\) for polynomials of the form NEWLINE\[NEWLINEf(x)=p^w+p^m\gamma_1x+\sum_{i=2}^d\gamma_ix^i,\;m\geq1, w\geq 2, d\geq 2,NEWLINE\]NEWLINE where \(\gamma_1,\dots,\gamma_d\in\mathbb{Z}\) and gcd(\(\gamma_1,p\))=1. Moreover the authors give a nice version of Hensel's lemma in \(\mathbb{Z}_p[[x]].\)