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 ![F_r[T]](/images/equations/Non-ArchimedeanValuation/Inline21.svg)
(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.
