Hensel's Lemma - A Fair Application of Newton's Method and 'Double Induction'
Introduction
Let $F$ be a non-Archimedean local field, meaning that $F$ is complete under the metric induced by a non-Archimedean absolute value $|\cdot|$. Consider the ring of integers
and its unique prime (hence maximal) ideal
The residue field $k=\mathfrak{o}_F/\mathfrak{p}$ is finite because it is compact and discrete. For compactness notice that $\mathfrak{o}_F$ is compact, and the canonical projection $\mathfrak{o}_F \to k$ is open. For discreteness, notice that $\mathfrak{p}$ is open, connected and contains the unit.
Let $f \in \mathfrak{o}_F[x]$ be a polynomial. Hensel’s lemma states that, if $\overline{f} \in k[x]$, the reduction of $f$, has a simple root $a$ in $k$, then the root can be lifted to a root of $f$ in $\mathfrak{o}_F$ and hence $F$. This blog post is intended to offer a well-organised proof of this lemma.
To do this, we need to use Newton’s method of approximating roots of $f(x)=0$, something like
We know that $a_n \to \zeta$ where $f(\zeta)=0$ at a $A^{2^n}$ speed for some constant $A$, in calculus (do Walter Rudin’s exercise 5.25 of Principles of Mathematical Analysis if you are not familiar with it, I heartily recommend.). Now we will steal Newton’s method into...
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