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| In [[mathematics]], a '''local field''' is a special type of [[Field (mathematics)|field]] that is a [[locally compact]] [[topological field]] with respect to a [[Discrete space|non-discrete topology]].<ref>Page 20 of {{Harvnb|Weil|1995}}</ref>
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| Given such a field, an [[Absolute value (algebra)|absolute value]] can be defined on it. There are two basic types of local field: those in which the absolute value 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'''. Local fields arise naturally in [[number theory]] as [[Completion (metric space)|completions]] of [[global field]]s.
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| Every local field is [[isomorphic]] (as a topological field) to one of the following:
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| *Archimedean local fields ([[Characteristic (algebra)|characteristic]] zero): the [[real numbers]] '''R''', and the [[complex numbers]] '''C'''.
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| *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]]).
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| *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'').
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| There is an equivalent definition of non-archimedean local field: it is a field that is [[complete valued field|complete with respect to a discrete valuation]] and whose [[residue field]] is finite. However, some authors consider a more general notion, requiring only that the residue field be [[Perfect field|perfect]], not necessarily finite.<ref>See, for example, definition 1.4.6 of {{harvnb|Fesenko|Vostokov|2002}}</ref> This article uses the former definition.
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| ==Induced absolute value==
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| Given a locally compact topological field ''K'', an absolute value can be defined as follows. First, consider the [[Field (mathematics)#Related algebraic structures|additive group]] of the field. As a locally compact [[topological group]], it has a unique (up to positive scalar multiple) [[Haar measure]] μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of ''K''. Specifically, define |·| : ''K'' → '''R''' by<ref>Page 4 of {{Harvnb|Weil|1995}}</ref>
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| :<math>|a|:=\frac{\mu(aX)}{\mu(X)}</math>
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| for any [[measurable]] subset ''X'' of ''K'' (with 0 < μ(X) < ∞). This absolute value does not depend on ''X'' nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the [[numerator]] and the [[denominator]]).
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| Given such an absolute value on ''K'', a new [[Normed space#Topological structure|induced topology]] can be defined on ''K''. This topology is the same as the original topology.<ref>Corollary 1, page 5 of {{Harvnb|Weil|1995}}</ref> Explicitly, for a positive real number ''m'', define the subset ''B''<sub>m</sub> of ''K'' by
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| :<math>B_m:=\{ a\in K:|a|\leq m\}.</math>
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| Then, the ''B''<sub>m</sub> make up a [[neighbourhood basis]] of 0 in ''K''.
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| ==<span id="normalizedvaluation"></span>Non-archimedean local field theory==
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| For a non-archimedean local field ''F'' (with absolute value denoted by |·|), the following objects are important:
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| *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]];
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| *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'';
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| *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>;
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| *a [[principal ideal|generator]] ϖ of <math>\mathfrak{m}</math> called a '''[[uniformizer]]''' of ''F'';
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| *its residue field <math>k=\mathcal{O}/\mathfrak{m}</math> which is finite (since it is compact and [[Discrete space|discrete]]).
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| 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.
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| 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<ref>{{harvnb|Weil|1995|loc=chapter I, theorem 6}}</ref>
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| :<math>|a|=q^{-v(a)}.</math>
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| An equivalent 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.
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| ===Examples===
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| <ol>
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| <li> '''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, then ''v''(''u'' ''p''<sup>n</sup>) = ''n'' for the normalized valuation.
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| <li> '''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 prime ideal is (''T'') (i.e. the power series whose [[constant term]] is 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:
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| ::<math>v\left(\sum_{i=-m}^\infty a_iT^i\right) = -m</math> (where ''a''<sub>−''m''</sub> is non-zero).
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| <li> 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.
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| </ol>
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| ===<span id="higherunit"></span><span id="principalunit"></span>Higher unit groups===
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| The '''''n''<sup>th</sup> higher unit group''' of a non-archimedean local field ''F'' is
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| :<math>U^{(n)}=1+\mathfrak{m}^n=\left\{u\in\mathcal{O}^\times:u\equiv1\, (\mathrm{mod}\,\mathfrak{m}^n)\right\}</math>
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| for ''n'' ≥ 1. The group ''U''<sup>(1)</sub> 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>.
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| The higher unit groups provide a decreasing [[filtration (mathematics)|filtration]] of the unit group
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| :<math>\mathcal{O}^\times\supseteq U^{(1)}\supseteq U^{(2)}\supseteq\cdots</math>
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| whose [[quotient group|quotients]] are given by
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| :<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>
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| for ''n'' ≥ 1.<ref>{{harvnb|Neukirch|1999|loc=p. 122}}</ref> (Here "<math>\approx</math>" means a non-canonical isomorphism.)
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| ===Structure of the unit group===
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| The multiplicative group of non-zero elements of a non-archimedean local field ''F'' is isomorphic to
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| :<math>F^\times\cong(\varpi)\times\mu_{q-1}\times U^{(1)}</math>
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| 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]]:
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| *If ''F'' has positive characteristic ''p'', then
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| ::<math>F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/{(q-1)}\oplus\mathbf{Z}_p^\mathbf{N}</math>
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| :where '''N''' denotes the [[natural number]]s;
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| *If ''F'' has characteristic zero (i.e. it is a finite extension of '''Q'''<sub>''p''</sub> of degree ''d''), then
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| ::<math>F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/(q-1)\oplus\mathbf{Z}/p^a\oplus\mathbf{Z}_p^d</math>
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| :where ''a'' ≥ 0 is defined so that the group of ''p''-power roots of unity in ''F'' is <math>\mu_{p^a}</math>.<ref>{{harvnb|Neukirch|1999|loc=theorem II.5.7}}</ref>
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| == Higher dimensional local fields ==
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| {{main|Higher local field}}
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| It is natural to introduce non-archimedean local fields in a uniform geometric way 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 generalizations, a local field is sometimes called a ''one-dimensional local field''.
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| 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.<ref>Definition 1.4.6 of {{Harvnb|Fesenko|Vostokov|2002}}</ref> 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 [[quasi-finite field]],<ref>{{Harvnb|Serre|1995}}</ref> or a perfect field.
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| 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.
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| ==See also==
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| * [[Hasse principle]]
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| * [[Local class field theory]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{Citation
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| | last=Serre
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| | first=Jean-Pierre
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| | author-link=Jean-Pierre Serre
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| | title=[[Local Fields (book)|Local Fields]]
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| | year=1995
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| | place=Berlin, Heidelberg
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| | publisher=[[Springer-Verlag]]
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| | series=[[Graduate texts in mathematics]]
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| | volume=67
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| | isbn=0-387-90424-7
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| }}
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| * {{Citation
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| | last=Weil
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| | first=André
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| | author-link=André Weil
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| | title=Basic number theory
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| | year=1995
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| | place=Berlin, Heidelberg
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| | publisher=[[Springer-Verlag]]
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| | series=Classics in Mathematics
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| | isbn=3-540-58655-5
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| }}
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| * {{Citation
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| | last=Fesenko
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| | first=Ivan B.
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| | author-link=Ivan Fesenko
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| | last2=Vostokov
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| | first2=Sergei V.
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| | title=Local fields and their extensions
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| | publisher=[[American Mathematical Society]]
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| | location=Providence, RI
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| | year=2002
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| | series=Translations of Mathematical Monographs
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| | volume=121
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| | edition=Second
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| | isbn=978-0-8218-3259-2
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| | mr=1915966
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| }}
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| *{{Neukirch ANT}}
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| ==Further reading==
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| * [[A. Frohlich]], "Local fields", in [[J.W.S. Cassels]] and A. Frohlich (edd), ''Algebraic number theory'', [[Academic Press]], 1973. Chap.I
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| * Milne, James, [http://www.jmilne.org/math/CourseNotes/ant.html '''Algebraic Number Theory'''].
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| * Schikhoff, W.H. (1984) ''Ultrametric Calculus''
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| ==External links==
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| * {{springer|title=Local field|id=p/l060130}}
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| {{DEFAULTSORT:Local Field}}
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| [[Category:Field theory]]
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| [[Category:Algebraic number theory]]
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