Talk:Local field
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I changed the definition of discrete valuation so that it directly maps into the integers. Otherwise, we would have to say it maps to an ordered abelian group isomorphic as ordered abelian groups to Z in order to be able speak about the elements with negative valutation, and that seems to be overkill.
Regarding the definition: in http://planetmath.org/encyclopedia/LocalField.html they include R and C as local fields. Is our definition the standard one? http://mathworld.wolfram.com/LocalField.html has yet another definition, which seems redundant however. They also give a different list of examples. AxelBoldt 19:18 Nov 9, 2002 (UTC)
I think that's probably clearer anyway. Good catch on the Laurent/power series mistake also, that was rather a gaffe. R and C are also referred to as local fields in the real numbers article. It depends on the definition: when I saw that and your question I recalled having read a definition of local fields as a locally compact Hausdorff topological field and that was that. I'll do some poking around today and probably add some remarks about the ambiguity of the definition. Also added finite extensions of Qp and finite extensions of function fields over finite fields to the zoo. alodyne
If we go for the "locally compact" definition, we have to be careful however, since the maximal-compact-subring statement won't remain true.
Can every finite extension of the p-adics be turned into a local field? And if so, can it be done in a natural or unique way? AxelBoldt 00:02 Nov 11, 2002 (UTC)
Under most usual definitions, the valuation domain R of a valued field K is either R = { x in K : |x| <= 1 } given the Artin-style valuations, or R = { x in K : v(x) >= 0 } given the Krull-style valuations. In this article one has the inequality backwards. Since valuation hasn't been defined yet, it's not wrong (yet), but is probably confusing.
When you say that the valuation takes its values in the integers, that is the Krull-style, v(x), and leaves out the value infinity for v(0). The Artin-style would take its values in the set { p^n : n in Z } union {0}, where p is probably a positive prime integer, but occasionally set to other numbers. These are all nitpicks until a definition has been given. For instance sometimes v is only defined on the nonzero elements of K.