Editing Local field (section)

==Basic features of non-Archimedean local fields==

For a non-Archimedean local field ''F'' (with absolute value denoted by |·|), the following objects are important:
*its '''[[ring of integers]]''' <math>\mathcal{O} = \{a\in F: |a|\leq 1\}</math> which is a [[discrete valuation ring]], is the closed [[unit ball]] of ''F'', and is [[Compact space|compact]];
*the '''units''' in its ring of integers <math>\mathcal{O}^\times = \{a\in F: |a|= 1\}</math> which forms a [[Group (mathematics)|group]] and is the [[unit sphere]] of ''F'';
*the unique non-zero [[prime ideal]] <math>\mathfrak{m}</math> in its ring of integers which is its open unit ball <math>\{a\in F: |a|< 1\}</math>;
*a [[principal ideal|generator]] <math>\varpi</math> of <math>\mathfrak{m}</math> called a '''[[uniformizer]]''' of <math>F</math>;
*its residue field <math>k=\mathcal{O}/\mathfrak{m}</math> which is finite (since it is compact and  [[Discrete space|discrete]]).

Every non-zero element ''a'' of ''F'' can be written as ''a'' = ϖ<sup>''n''</sup>''u'' with ''u'' a unit, and ''n'' a unique integer.
The '''normalized valuation''' of ''F'' is the [[surjective function]] ''v'' : ''F'' → '''Z''' ∪ {∞} defined by sending a non-zero ''a'' to the unique integer ''n'' such that ''a'' = ϖ<sup>''n''</sup>''u'' with ''u'' a unit, and by sending 0 to ∞. If ''q'' is the [[cardinality]] of the residue field, the absolute value on ''F'' induced by its structure as a local field is given by:{{sfn|Weil|1995|loc=Ch. I, Theorem 6}}
:<math>|a|=q^{-v(a)}.</math>

An equivalent and very important definition of a non-Archimedean local field is that it is a field that is [[complete valued field|complete with respect to a discrete valuation]] and whose residue field is finite.

===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.

===<span id="higherunit"></span><span id="principalunit"></span>Higher unit groups===

The '''''n''<sup>th</sup> higher unit group''' of a non-Archimedean local field ''F'' is
:<math>U^{(n)}=1+\mathfrak{m}^n=\left\{u\in\mathcal{O}^\times:u\equiv1\, (\mathrm{mod}\,\mathfrak{m}^n)\right\}</math>
for ''n''&nbsp;≥&nbsp;1. The group ''U''<sup>(1)</sup> is called the '''group of principal units''', and any element of it is called a '''principal unit'''. The full unit group <math>\mathcal{O}^\times</math> is denoted ''U''<sup>(0)</sup>.

The higher unit groups form a decreasing [[filtration (mathematics)|filtration]] of the unit group
:<math>\mathcal{O}^\times\supseteq U^{(1)}\supseteq U^{(2)}\supseteq\cdots</math>
whose [[quotient group|quotients]] are given by
:<math>\mathcal{O}^\times/U^{(n)}\cong\left(\mathcal{O}/\mathfrak{m}^n\right)^\times\text{ and }\,U^{(n)}/U^{(n+1)}\approx\mathcal{O}/\mathfrak{m}</math>
for ''n''&nbsp;≥&nbsp;1.{{sfn|Neukirch|1999|p=122}} (Here "<math>\approx</math>" means a non-canonical isomorphism.)

===Structure of the unit group===

The multiplicative group of non-zero elements of a non-Archimedean local field ''F'' is isomorphic to
:<math>F^\times\cong(\varpi)\times\mu_{q-1}\times U^{(1)}</math>
where ''q'' is the order of the residue field, and μ<sub>''q''−1</sub> is the group of (''q''−1)st roots of unity (in ''F''). Its structure as an abelian group depends on its [[characteristic (algebra)|characteristic]]:
*If ''F'' has positive characteristic ''p'', then
::<math>F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/{(q-1)}\oplus\mathbf{Z}_p^\mathbf{N}</math>
:where '''N''' denotes the [[natural number]]s;
*If ''F'' has characteristic zero (i.e. it is a finite extension of '''Q'''<sub>''p''</sub> of degree ''d''), then
::<math>F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/(q-1)\oplus\mathbf{Z}/p^a\oplus\mathbf{Z}_p^d</math>
:where ''a''&nbsp;≥&nbsp;0 is defined so that the group of ''p''-power roots of unity in ''F'' is <math>\mu_{p^a}</math>.{{sfn|Neukirch|1999|loc=Theorem II.5.7}}