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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>
| | If you have the desire to procedure settings quickly, loading files promptly, but the body is logy and torpid, what would we do? If you are a giant "switchboard" which is lack of efficient management program plus effective housekeeper, what would you do? If you have send a exact commands to the notice, nevertheless the body cannot work properly, what would we do? Yes! We require a full-featured repair registry!<br><br>You are able to reformat a computer to make it run faster. This may reset the computer to whenever we initially utilized it. Always remember to back up all files and programs before doing this since this will remove your files from the database. Remember before we do this we need all of the motorists and installation files plus this should be a last resort in the event you are interested in slow computer strategies.<br><br>Registry cleaning is important because the registry will get crowded plus messy whenever it is very left unchecked. False entries send the running program interested in files and directories that have long ago been deleted. This takes time plus uses valuable resources. So, a slowdown inevitably happens. It is specifically noticeable when we multitask.<br><br>Windows errors is caused by any number of reasons, however there's virtually constantly one cause. There's a hidden piece of your system that is responsible for creating 90% of all Windows errors, plus it's called the 'registry'. This is the central database for a system plus is where a computer stores all its program files plus settings. It's a extremely important part of Windows, that is should be able to function. However, it's equally among the biggest causes of problems on a PC.<br><br>Besides, in the event you will receive a [http://bestregistrycleanerfix.com/registry-mechanic pc tools registry mechanic] which may work for you effectively plus rapidly, then why not? There is one such system, RegCure that is great and complete. It has attributes that alternative products never have. It is the many recommended registry cleaner today.<br><br>Why this issue occurs frequently? What are the causes of it? In truth, there are 3 major causes that will lead to the PC freezing problem. To solve the problem, we have to take 3 steps inside the following paragraphs.<br><br>To accelerate your computer, you simply should be capable to get rid of all these junk files, permitting a computer to find what it wants, when it wants. Luckily, there's a tool which allows you to do this conveniently and fast. It's a tool called a 'registry cleaner'.<br><br>You are able to click here to find out how to speed up Windows plus increase PC perfomance. And you can click here to download a registry cleaner to aid we clean up registry. |
| 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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