Let
be a field of arbitrary characteristic. Let be defined by the following properties:

1. ,

2. ,
and

3. .

Then
is said to be a non-Archimedean valuation and is said to be a non-Archimedean valued field. For example,
for ,
if ,
can be decomposed as where and neither the numerator nor denominator of involve . Then is a non-Archimedean valuation.

Another example can be constructed by letting be the field of formal Laurent series over the finite
field with
elements ,
taking
the quotient field of (the polynomial ring in the variable over ), and setting . If is written with the numerator and denominator of relatively prime to , then is a non-Archimedean valuation.

Let
be a non-Archimedean valuation and let with . The non-Archimedean absolute value is obtained by setting .

has the following properties:

1.

2.

3.
(non-Archimedean triangular inequality).

An absolute value over a field is non-Archimedean iff for all .

The completion of is the completion of the metric space . In the example above, is the completion of with respect to the valuation .

Non-Archimedean complete fields satisfy the following properties:

1.
converges .
(Note that for Archimedean valuations we only have implication.)

2. If ,
then
converges to a nonzero element if and only if .

Let

so
is the valuation ring of and

Then
is a local ring and is its maximal ideal. If is a non-Archimedean complete field, then is compact and is a finite field. Suppose that the cardinality of
is ,
then the definition can be made and the absolute value is said to be normalized.

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Alden-Biesen, 9-14 September 1996 (Ed. E.-U. Gekeler, M. van der Put, M. Reversat,
and J. Van Geel). Singapore: World Scientific, pp. 32-33, 1997.Goss,
D. Basic
Structures of Function Field Arithmetic. Berlin: Springer-Verlag, pp. 35-45,
1996.Jacobson, N. Basic
Algebra II, 2nd ed. New York: W. H. Freeman, pp. 557-618,
1989.