On the Hensel lift of a polynomial

Denote by R the Galois ring of characteristic p^e and cardinality p^em, where p is a prime and e and m are positive integers. Let g(x) be a monic polynomial over F_pm. A polynomial f(x) over R is defined to be a Hensel lift of g(x) in R[x] if f¯(x)=g(x), and there is a positive integer n not divisible by p such that f(x) divides xⁿ-1 in R[x]. It is proved that g(x) has a unique Hensel lift in R[x] if and only if g(x) has no multiple roots and xχg(x). An algorithm to compute the Hensel lift is also given