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 $\mathrm{|}\mathrm{\cdot }\mathrm{|}$. Let ${\mathrm{O}}_K$ be the ring of integers in $K$ (i.e. the set of elements of $K$ with $\mathrm{|}k\mathrm{|}\mathrm{\le }\mathrm{1}$). Let $f\mathit{}\mathrm{(}x\mathrm{)}$ be a polynomial with coefficients in ${\mathrm{O}}_K$ and suppose there exist ${\alpha }_{\mathrm{0}}\mathrm{\in }{\mathrm{O}}_K$ such that

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

$${\alpha }_{i+1}={\alpha }_i-\frac{f({\alpha }_i)}{f^{\prime }({\alpha }_i)}$$

converges to $\alpha$. Furthermore:

Corollary (Trivial case of Hensel’s lemma).

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

$$f({\alpha }_0)\equiv 0mod𝔭,f^{\prime }({\alpha }_0)\ne 0mod𝔭.$$

Then there exist a root $\alpha \mathrm{\in }{K}_{\mathrm{p}}$ of $f\mathit{}\mathrm{(}x\mathrm{)}$, i.e. $f\mathit{}\mathrm{(}\alpha \mathrm{)}\mathrm{=}\mathrm{0}$.