Editing Local field (section)

===Examples===

#'''The ''p''-adic numbers''': the ring of integers of '''Q'''<sub>''p''</sub> is the ring of ''p''-adic integers '''Z'''<sub>''p''</sub>. Its prime ideal is ''p'''''Z'''<sub>''p''</sub> and its residue field is '''Z'''/''p'''''Z'''. Every non-zero element of '''Q'''<sub>p</sub> can be written as ''u'' ''p''<sup>''n''</sup> where ''u'' is a unit in '''Z'''<sub>''p''</sub> and ''n'' is an integer, with ''v''(''u'' ''p''<sup>n</sup>) = ''n'' for the normalized valuation.
#'''The formal Laurent series over a finite field''': the ring of integers of '''F'''<sub>''q''</sub>((''T'')) is the ring of [[formal power series]] '''F'''<sub>''q''</sub><nowiki>[[</nowiki>''T''<nowiki>]]</nowiki>. Its maximal ideal is (''T'') (i.e. the set of [[power series]] whose [[constant term]]s are zero) and its residue field is '''F'''<sub>''q''</sub>. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:
#::<math>v\left(\sum_{i=-m}^\infty a_iT^i\right) = -m</math> (where ''a''<sub>&minus;''m''</sub> is non-zero).
#The formal Laurent series over the complex numbers is ''not'' a local field. For example, its residue field is '''C'''<nowiki>[[</nowiki>''T''<nowiki>]]</nowiki>/(''T'') = '''C''', which is not finite.