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A field equipped with a Krull valuation or an absolute value.
A general Krull valuation takes values in an ordered abelian group and canonically induces a topology/uniformity. A real-valued metric is available for absolute values and rank-one valuations, not for every value group without additional choices.
wp-valued-fieldmilne-algebraic-number-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Equip the field with a valuation or absolute value.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-valued-fieldmilne-algebraic-number-theoryEach relation below starts at this concept.
This 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-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Cauchy completion produces a complete valued field containing a dense image of the original field.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
milne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
A valued field includes an underlying field together with its chosen valuation or absolute value; forgetting additional topological or completeness properties retains that valuation structure.
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.
wp-valued-fieldmilne-algebraic-number-theory