info prev up next book cdrom email home

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,
\end{displaymath} (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.
\end{displaymath} (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,
\end{displaymath} (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}}.
\end{displaymath} (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}}.
\end{displaymath} (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.

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein