discrete valuation
A discrete valuation on a field is a valuation whose image is a discrete subset of .
For any field with a discrete valuation , the set
is a subring of with sole maximal ideal
and hence is a discrete valuation ring. Conversely, given any discrete valuation ring , the field of fractions of admits a discrete valuation sending each element to , where is some arbitrary fixed constant and is the order of , and extending multiplicatively to .
Note: Discrete valuations are often written additively instead of multiplicatively; under this alternate viewpoint, the element maps to (in the above notation) instead of just . This transformation reverses the order of the absolute values (since ), and sends the element to . It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the integers.
Title | discrete valuation |
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Canonical name | DiscreteValuation |
Date of creation | 2013-03-22 13:59:14 |
Last modified on | 2013-03-22 13:59:14 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 6 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 13F30 |
Classification | msc 12J20 |
Synonym | rank one valuations |
Synonym | rank-one valuations |
Related topic | DiscreteValuationRing |
Related topic | Valuation |