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A map from a field or integral domain to an ordered abelian group with satisfying and .
Equivalent multiplicative conventions use a non-Archimedean absolute value; Archimedean absolute values are treated separately when discussing places of number fields.
wp-valuationnlab-valuationmilne-algebraic-number-theoryEach relation below ends at this concept.
This 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-theoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
For a valuation on a field, the elements of nonnegative valuation form its valuation ring; equivalent multiplicative conventions select the elements of value at most one.
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-theory