I am reading the textbook "Finite Fields and Galois Rings" by Wan, and am confused by the definition of a Hensel lift and how it relates to Hensel's Lemma.
The result that is labelled Hensel's Lemma (Lemma 13.7) is the existence, for any factorization of a polynomial in $\mathbb{F}_p[x]$ into monic, coprime polynomials, of an analog factorization in $\mathbb{Z}_{p^s}[x]$, where each factor is the preimage of a factor downstairs. This seems to be what is required in order to "lift" a solution to a polynomial equation from the field to the Galois ring, which is what I thought Hensel lifting was about.
Later (Definition 13.1), a Hensel lift of a monic polynomial over $\mathbb{F}_p$ is defined as a polynomial as a preimage polynomial over the Galois ring that divides $x^n - 1$ (in the ring of polynomials over the Galois ring) for a positive integer $n$ not dividing $p$. Is this a standard definition of a Hensel lift? Why does it concern lifting individual polynomials, instead of lifting solutions or lifting factorizations? And where did the condition involving $x^n - 1$ come from?
Any help understanding would be greatly appreciated.
