Hensel’s lemma

The following results are used to show the existence of a solution to polynomial equations over local fields. Notice the similarities with Newton’s method.

Theorem (Hensel’s Lemma).

Let $K$ be a local field (http://planetmath.org/LocalField), complete with respect to a valuation $|\cdot|$. Let $\mathcal{O}_{K}$ be the ring of integers in $K$ (i.e. the set of elements of $K$ with $|k|\leq 1$). Let $f(x)$ be a polynomial with coefficients in $\mathcal{O}_{K}$ and suppose there exist $\alpha_{0}\in\mathcal{O}_{K}$ such that

 $|f(\alpha_{0})|<|f^{\prime}(\alpha_{0})^{2}|.$

Then there exist a root $\alpha\in K$ of $f(x)$. Moreover, the sequence:

 $\alpha_{i+1}=\alpha_{i}-\frac{f(\alpha_{i})}{f^{\prime}(\alpha_{i})}$

converges to $\alpha$. Furthermore:

 $|\alpha-\alpha_{0}|\leq\left|\frac{f(\alpha_{i})}{f^{\prime}(\alpha_{i})}% \right|<1.$
Corollary (Trivial case of Hensel’s lemma).

Let $K$ be a number field and let $\mathfrak{p}$ be a prime ideal in the ring of integers $\mathcal{O}_{K}$. Let $K_{\mathfrak{p}}$ be the completion of $K$ at the finite place $\mathfrak{p}$ and let $\mathcal{O}_{\mathfrak{p}}$ be the ring of integers in $K_{\mathfrak{p}}$. Let $f(x)$ be a polynomial with coefficients in $\mathcal{O}_{\mathfrak{p}}$ and suppose there exist $\alpha_{0}\in\mathcal{O}_{\mathfrak{p}}$ such that

 $f(\alpha_{0})\equiv 0\mod\mathfrak{p},\quad f^{\prime}(\alpha_{0})\neq 0\mod% \mathfrak{p}.$

Then there exist a root $\alpha\in K_{\mathfrak{p}}$ of $f(x)$, i.e. $f(\alpha)=0$.

Title Hensel’s lemma HenselsLemma 2013-03-22 15:08:30 2013-03-22 15:08:30 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 13H99 msc 12J99 msc 11S99