 TOPICS # Non-Archimedean Valuation

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.

Krasner's Lemma, Non-Archimedean Field, Non-Archimedean Geometry

This entry contributed by José Gallardo Alberni

## Explore with Wolfram|Alpha ## References

López, B. "Analytic Theory of Drinfeld Modules." Proceedings of the Workshop on Drinfeld Modules, Modular Schemes and Applications, 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.

## Referenced on Wolfram|Alpha

Non-Archimedean Valuation

## Cite this as:

Alberni, José Gallardo. "Non-Archimedean Valuation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Non-ArchimedeanValuation.html