A non-Archimedean valuation on a field 
of arbitrary characteristic
is a map 
satisfying the following properties:
1. 
,
2. 
,
and
3. 
.
The pair 
is 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 is the field 
of formal Laurent
series in 
over the finite field 
with 
elements. For a nonzero series, 
is the least exponent of 
with nonzero coefficient, and 
. On its rational-function subfield 
, a nonzero element 
has 
.
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 with distance 
. 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 iff 
.
Let
 |
so 
is the valuation ring of 
and
 |
Then 
is a local ring and 
is its maximal ideal. If
is complete for a nontrivial discrete valuation, then 
is compact iff the residue field 
is finite (Milne 2020, Proposition 7.46). Completeness
alone does not imply either condition. For a discrete valuation normalized by 
and a residue field of cardinality 
, the choice 
gives the normalized absolute
value.
