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 (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
 |
(2)
|
 |
(3)
|
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
 |
(4)
|
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
 |
(5)
|
The 
-adic valuation on 
is discrete, but the ordinary absolute valuation is not.
If 
is a valuation on 
, then it induces a metric
 |
(6)
|
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.
