For any field with a discrete valuation , the set
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.
|Date of creation||2013-03-22 13:59:14|
|Last modified on||2013-03-22 13:59:14|
|Last modified by||djao (24)|
|Synonym||rank one valuations|