Hensel's Lemma

An important result in Valuation Theory which gives information on finding roots of Polynomials. Hensel's lemma is formally stated as follow. Let $(K, \vert\cdot\vert)$ be a complete non-Archimedean valuated field, and let $R$ be the corresponding Valuation Ring. Let $f(x)$ be a Polynomial whose Coefficients are in $R$ and suppose $a_0$ satisfies

$$\vert f(a_0)\vert < \vert f'(a_0)\vert^2,$$ (1)

where $f'$ is the (formal) Derivative of $f$. Then there exists a unique element $a\in R$ such that $f(a)=0$ and

$$\vert a-a_0\vert \leq \left\vert{f(a_0)\over f'(a_0)}\right\vert.$$ (2)

Less formally, if $f(x)$ is a Polynomial with ``Integer'' Coefficients and $f(a_0)$ is ``small'' compared to $f'(a_0)$, then the equation $f(x)=0$ has a solution ``near'' $a_0$. In addition, there are no other solutions near $a_0$, although there may be other solutions. The proof of the Lemma is based around the Newton-Raphson method and relies on the non-Archimedean nature of the valuation.

Consider the following example in which Hensel's lemma is used to determine that the equation $x^2 = -1$ is solvable in the 5-adic numbers $\Bbb{Q}_5$ (and so we can embed the Gaussian Integers inside $\Bbb{Q}_5$ in a nice way). Let $K$ be the 5-adic numbers $\Bbb{Q}_5$, let $f(x)=x^2+1$, and let $a_0 = 2$. Then we have $f(2) = 5$ and $f'(2) = 4$, so

$$\vert f(2)\vert _5 = {\textstyle{1\over 5}} < \vert f'(2)\vert _5^2 = 1,$$ (3)

and the condition is satisfied. Hensel's lemma then tells us that there is a 5-adic number $a$ such that $a^2 + 1 = 0$ and

$$\vert a - 2\vert _5 <= \vert{\textstyle{5\over 4}}\vert _5 = {\textstyle{1\over 5}}.$$ (4)

Similarly, there is a 5-adic number $b$ such that $b^2 + 1 = 0$ and

$$\vert b - 3\vert _5 <= \vert{\textstyle{10\over 7}}\vert _5 = {\textstyle{1\over 5}}.$$ (5)

Therefore, we have found both the square roots of $-1$ in $\Bbb{Q}_5$. It is possible to find the roots of any Polynomial using this technique.