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In the question here, the subject of "Analysis in Positive Characteristic" is mentioned. Looking at Wikipedia's local field, this is the final type of analysis in local fields to be developed following on from real, complex and p-adic analysis. ( Is there a snappy name for this? Charpos analysis ? ff/Fq analysis ? Carlitz analysis ?)

Question: What is the situation with regard to division rings - Is there a classification of local division rings? and for which ones has analysis been developed ?

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The classification of locally compact (non necessarily commutative) fields is done in chapter 1 of André Weil's Basic Number Theory. His method is very nice, based on the existence of a Haar measure on a locally compact group.

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For a more concrete approach to the classification of (nondiscrete) locally compact division rings, see Jacobson, Basic Algebra II. –  KConrad Dec 30 '11 at 6:31
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