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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?

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    The examples given after the proof in that link are not terribly compelling. His version of Hensel's lemma involves the lifting of fairly general factors over A/m to a factorization over A but he only illustrates it with simple linear factors, for which the simpler version of Hensel's lemma about lifting simple roots would be adequate.2011-09-08

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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 $< r$.

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    Sure, that's why I said in point 2 that "\delta" is not really fixed as in the notes". I'm not sure how the author proceeded though.2011-09-08