Is there a topological characterisation of non-Archimedean local fields? A local field is a locally compact field with a non-discrete topology. They classify as:


*

*Archimedean, Char=0 : The Real line, or the Complex plane

*Non-Archimedean, Char=0: Finite extensions of the p-adic rationals

*Non-Archimedean, Char=p: Laurent series over a finite field
This is shown by the natural absolute value built from field by using the Haar measure (which is unique) for the additive structure, that is $|a|:=\mu(aK)/\mu(K)$ for any set $K$ of finite measure. This is well-defined since scalar factors cancel.
Is there a topological characterisation of the Archimedean property here? The characterisation I've seen in wikipedia uses the ring structure:

it is a field that is complete with respect to a discrete valuation and whose residue field is finite.


Archimedian local fields are connected, non-archimedian local fields are totally disconnected.
