Editing Local field

{{Short description|Locally compact topological field}}
In [[mathematics]], a [[Field (mathematics)|field]] ''K'' is called a non-Archimedean '''local field''' if it is [[Complete metric space|complete]] with respect to a [[Metric space|metric]] induced by a [[discrete valuation]] ''v'' and if its [[residue field]] ''k'' is finite.{{sfn|Cassels|Fröhlich|1967|loc=Ch. VI, Intro.|p=129}} In general, a local field is a [[locally compact]] [[topological field]] with respect to a [[Discrete space|non-discrete topology]].{{sfn|Weil|1995|p=20}} The [[real numbers]] '''R''', and the [[complex numbers]] '''C''' (with their standard topologies) are Archimedean local fields. Given a local field, the [[Valuation (algebra)|valuation]] defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is [[Archimedean property|Archimedean]] and those in which it is not. In the first case, one calls the local field an '''Archimedean local field''', in the second case, one calls it a '''non-Archimedean local field'''.{{sfn|Milne|2020|loc=Remark 7.49|p=127}} Local fields arise naturally in [[number theory]] as completions of [[global field]]s.{{sfn|Neukirch|1999|loc=Sec. 5|p=134}}

While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local fields, the fields of [[p-adic number|''p''-adic number]]s for  positive prime integer ''p'', were introduced by [[Kurt Hensel]]  at the end of the 19th century.

Every local field is [[isomorphic]] (as a topological field) to one of the following:{{sfn|Milne|2020|loc=Remark 7.49|p=127}}

*Archimedean local fields ([[Characteristic (algebra)|characteristic]] zero): the [[real numbers]] '''R''', and the [[complex numbers]] '''C'''.
*Non-Archimedean local fields of characteristic zero: [[finite extension]]s of the [[p-adic number|''p''-adic number]]s '''Q'''<sub>''p''</sub> (where ''p'' is any [[prime number]]).
*Non-Archimedean local fields of characteristic ''p'' (for ''p'' any given prime number): the field of [[formal Laurent series]] '''F'''<sub>''q''</sub>((''T'')) over a [[finite field]] '''F'''<sub>''q''</sub>, where ''q'' is a [[Exponentiation|power]] of ''p''.

In particular, of importance in number theory, classes of local fields show up as the completions of [[algebraic number field]]s with respect to their discrete valuation corresponding to one of their [[maximal ideal]]s. Research papers in modern number theory often consider a more general notion, requiring only that the residue field be [[Perfect field|perfect]] of positive characteristic, not necessarily finite.{{sfn|Fesenko|Vostokov|2002|loc=Def. 1.4.6}} This article uses the former definition.

==Induced absolute value==

Given such an absolute value on a field ''K'', the following topology can be defined on ''K'': for a positive real number ''m'', define the subset ''B''<sub>m</sub> of ''K'' by
:<math>B_m:=\{ a\in K:|a|\leq m\}.</math>
Then, the ''b+B''<sub>m</sub> make up a [[neighbourhood basis]] of b in ''K''.

Conversely, a topological field with a non-discrete locally compact topology has an absolute value defining its topology. It can be constructed using the [[Haar measure]] of the [[Field (mathematics)#Related algebraic structures|additive group]] of the field.

==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}}

== Theory of local fields ==
This theory includes the study of types of local fields, extensions of local fields using [[Hensel's lemma]], [[Galois extension]]s of local fields, [[ramification group]]s filtrations of [[Galois group]]s of local fields, the behavior of the norm map on local fields, the local reciprocity homomorphism and existence theorem in [[local class field theory]], [[local Langlands correspondence]], [[Hodge-Tate theory]] (also called [[p-adic Hodge theory|''p''-adic Hodge theory]]), explicit formulas for the [[Hilbert symbol]] in local class field theory, see e.g.{{sfn|Fesenko|Vostokov|2002|loc=Chapters 1-4, 7}}

== Higher-dimensional local fields ==
{{main|Higher local field}}
A local field is sometimes called a ''one-dimensional local field''.

A non-Archimedean local field can be viewed as the field of fractions of the completion of the [[local ring]] of a one-dimensional arithmetic scheme of rank 1 at its non-singular point.

For a [[non-negative integer]] ''n'', an ''n''-dimensional local field is a complete discrete valuation field whose residue field is an  (''n'' − 1)-dimensional local field.{{sfn|Fesenko|Vostokov|2002|loc=Def. 1.4.6}} Depending on the definition of local field, a ''zero-dimensional local field'' is then either a finite field (with the definition used in this article), or a perfect field of positive characteristic.

From the geometric point of view, ''n''-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an ''n''-dimensional arithmetic scheme.

==See also==
* [[Hensel's lemma]]
* [[Ramification group]]
* [[Local class field theory]]
* [[Higher local field]]

== Citations ==
{{Reflist}}

==References==
{{refbegin}}
*{{Citation|publisher=[[Academic Press]] | editor1-first=J.W.S. | editor1-last=Cassels | editor1-link=J. W. S. Cassels | editor2-first=Albrecht | editor2-last= Fröhlich | editor2-link=Albrecht Fröhlich | title=Algebraic Number Theory | year=1967 | zbl=0153.07403}}
* {{Citation
| last1=Fesenko
| first1=Ivan B.
| author-link=Ivan Fesenko
| last2=Vostokov
| first2=Sergei V.
| title=Local fields and their extensions
| publisher=[[American Mathematical Society]]
| location=Providence, RI
| year=2002
| series=Translations of Mathematical Monographs
| volume=121
| edition=Second
| isbn=978-0-8218-3259-2
| mr=1915966 
}}
*{{Citation | last1=Milne | first1=James S. | author-link=James S. Milne | title=Algebraic Number Theory | url=https://www.jmilne.org/math/CourseNotes/ant.html| year=2020 | edition=3.08 }}
*{{Neukirch ANT|trans=true}}
* {{Citation
| last=Weil
| first=André
| author-link=André Weil
| title=Basic number theory
| year=1995
| place=Berlin, Heidelberg
| publisher=[[Springer-Verlag]]
| series=Classics in Mathematics
| isbn=3-540-58655-5
}}
* {{Citation
| last=Serre
| first=Jean-Pierre
| author-link=Jean-Pierre Serre
| title=Local Fields
| publisher=Springer-Verlag
| location=New York
| year=1979
| series=Graduate Texts in Mathematics
| volume=67
| edition=First
| isbn=0-387-90424-7
}}
{{refend}}

==External links==
* {{springer|title=Local field|id=p/l060130}}

{{Authority control}}

{{DEFAULTSORT:Local Field}}
[[Category:Field (mathematics)]]
[[Category:Algebraic number theory]]