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 :
2. iff ,
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
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
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 (4a). In view of this, we need only to study valuations satisfying (4a), and we often view axioms (4) and (4a) as interchangeable (although this is not strictly true).
If two valuations are equivalent, then they are both non-Archimedean or both Archimedean. , , and with the usual Euclidean norms are Archimedean valuated fields. For any prime , the p-adic numbers with the -adic valuation is a non-Archimedean field.
If is any field, we can define the trivial valuation on by for all and , which is a non-Archimedean valuation. If is a finite field, then the only possible valuation over is the trivial one. It can be shown that any valuation on is equivalent to one of the following: the trivial valuation, Euclidean absolute norm , or -adic valuation .
The equivalence of any nontrivial valuation of to either the usual absolute value or to a p-adic norm was proved by Ostrowski (1935). Equivalent valuations give rise to the same topology. Conversely, if two valuations have the same topology, then they are equivalent. A stronger result is the following: Let , , ..., be valuations over which are pairwise inequivalent and let , , ..., be elements of . Then there exists an infinite sequence (, , ...) of elements of such that
etc. This says that inequivalent valuations are, in some sense, completely independent of each other. For example, consider the rationals with the 3-adic and 5-adic valuations and , and consider the sequence of numbers given by
Then as with respect to , but as with respect to , illustrating that a sequence of numbers can tend to two different limits under two different valuations.
A discrete valuation is a valuation for which the valuation group is a discrete subset of the real numbers . Equivalently, a valuation (on a field ) is discrete if there exists a real number such that
The -adic valuation on is discrete, but the ordinary absolute valuation is not.
If is a valuation on , then it induces a metric
on , which in turn induces a topology on . If satisfies (4b), then the metric is an ultrametric. We say that is a complete valuated field if the metric space is complete.