A generalization of the p-adic norm first proposed by Kürschák in 1913. A valuation on a field is a function from to the real numbers such that the following properties hold for all : 1. , 2. iff , 3. , 4. implies for some constant (independent of ). If (4) is satisfied for , then satisfies the triangle inequality, 4a. for all . If (4) is satisfied for then satisfies the stronger ultrametric inequality 4b. . The simplest valuation is the absolute value for real numbers. A valuation satisfying (4b) is called non-Archimedean valuation; otherwise, it is called Archimedean.If is a valuation on and , then we can define a new valuation by(1)This does indeed give a valuation, but possibly with a different constant in axiom 4. If two valuations are related in this way, they are said to be equivalent, and this gives an equivalence relation on the collection of all valuations on . Any valuation is equivalent to one which satisfies the triangle inequality..
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..
A valuation for which implies for the constant (independent of ). Such a valuation does not satisfy the strong triangle inequality
Let be a non-Archimedean field. Its valuation ring is defined to beThe valuation ring has maximal idealand the field is called the residue field, class field, or field of digits. For example, if (p-adic numbers), then (-adic integers), (-adic integers congruent to 0 mod ), and = GF(), the finite field of order .