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