Completeness

A

For example, (usual archimedean absolute value) defines a metric on . The completion of with respect to this metric is the field of real numbers. More generally, whenever is a valuation on a field that satisfies the triangle inequality, then defines a metric on . Consider for the rest of this section only valuations that satisfy the triangle inequality.

w.r.t.

(i.e.,
).

To see that is unique up to a unique isomorphism fixing , we observe that there are no nontrivial continuous automorphisms that fix . This is because, by denseness, a continuous automorphism is determined by what it does to , and by assumption is the identity map on . More precisely, suppose and is a positive integer. Then by continuity there is (with ) such that if and then . Since is dense in , we can choose the above to be an element of . Then by hypothesis , so . Thus .

For the second, suppose that is non-archimedean (but not necessarily discrete). Suppose with . First I claim that there is such that . To see this, let , where is some element of with , note that , and choose such that , so