TITLE: Proof of Hensel's lemma
QUESTION [5 upvotes]: I am reading up the proof of Hensel's lemma here.
On page 2, after equation 2, the author concludes that the degree of $\delta h_k$ is less than $n$ since the degree of $\Delta$ and $\epsilon g_k$ is less than $n$. I am not sure I understand this. We only know that $\Delta \cong  \epsilon g_k +\delta h_k \mod{\mathfrak{m}^{k+1}[x]}$, so how do we get this equality of degrees?

REPLY [2 votes]: I'm not sure what he has in mind, but one way to proceed is to apply division algorithm to $\Delta - \epsilon g_k$ divided by $h_k$ in $(A/m^{k+1})[x]$ in equation 2. Equation 2 tells you it's divisible and uniqueness tells you $\delta$ has degree $< r$.