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A field complete for the uniformity induced by its valuation; in the absolute-value/rank-one case, complete for the associated metric.
The metric wording is restricted to absolute values or rank-one valuations. General valued fields are naturally uniform spaces.
wp-complete-valuednlab-complete-fieldmilne-algebraic-number-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Completing a valued field produces a complete valued field into which the original field maps densely; the completion is unique up to unique isometric isomorphism over the original field.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
milne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Require completeness for the valuation-induced uniformity; for absolute values or rank-one valuations this is the familiar metric completeness condition.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-complete-valuednlab-complete-fieldmilne-algebraic-number-theoryEach relation below starts at this concept.
No direct relations are authored in this direction.