An important result in valuation theory which gives information on finding roots of polynomials.
Hensel's lemma is formally stated as follows. Let 
be a complete non-Archimedean
field, and let 
be the corresponding valuation ring. Let 
be a polynomial whose coefficients are in 
and suppose 
satisfies
 |
(1)
|
where 
is the (formal) derivative of 
. Then there exists a unique element 
such that 
and
 |
(2)
|
Less formally, if 
is a polynomial with "integer"
coefficients and 
is "small" compared to 
, then the equation 
has a solution "near" 
. In addition, there are no other solutions near 
, 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 
is solvable in the 5-adic numbers 
(and so we can embed the Gaussian
integers inside 
in a nice way). Let 
be the 5-adic numbers 
,
let 
,
and let 
.
Then we have 
and 
,
so
 |
(3)
|
and the condition is satisfied. Hensel's lemma then tells us that there is a 5-adic number 
such that 
and
 |
(4)
|
Similarly, there is a 5-adic number 
such that 
and
 |
(5)
|
Therefore, we have found both the square roots of 
in 
. It is possible to find the roots of any polynomial
using this technique.
