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Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field? I understand that a non-Archimedean field is a field which does not satisfy the Archimedean property.

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Can you include your definition of "non-archimedean field"? – Matt Aug 24 '12 at 14:31
Usually, "non-archimedean" refers to the valuation. Hence, I don't see how a (non-)archimedean field could be so without it being also a valuation field. – M Turgeon Aug 24 '12 at 14:34
@MTurgeon: Perhaps iago is thinking of non-Archimedean ordered fields, like the hyperreals and surreals. – Brian M. Scott Aug 24 '12 at 14:39
A non-archimedian ordered field has a natural valuation on it. (Of course the values of that valuation need not be real numbers, but belong to some ordered abelian group.) – GEdgar Aug 24 '12 at 15:57
For example, N. Alling has shown that the surreal numbers are isomorphic (as an ordered and valued field) to the field of Hahn series with real coefficients and value group the surreal numbers themselves. – Brian M. Scott Aug 24 '12 at 16:06

Non-archimedean norms on a field are in bijection with with valuations via $v(a) = -\log |a|$. This is used in the connection between tropical geometry and non-archimedian amoebas. See this paper:

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