Local field
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*Non-Archimedean local fields of characteristic ''p'' (for ''p'' any given prime number): the field '''F'''''q''((''T'')) of [[formal Laurent series]] in the variable ''T'' over a [[finite field]] '''F'''''q'', where ''q'' is a [[Exponentiation|power]] of ''p''. |
*Non-Archimedean local fields of characteristic ''p'' (for ''p'' any given prime number): the field '''F'''''q''((''T'')) of [[formal Laurent series]] in the variable ''T'' over a [[finite field]] '''F'''''q'', where ''q'' is a [[Exponentiation|power]] of ''p''. |
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==Module, absolute value, metric== |
== Module, absolute value, metric == |
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Given a local field ''F'', a "module function" on ''F'' 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 module of an element ''a'' of ''F'' is defined so as to measure the change in size of a set after multiplying it by ''a''. Specifically, define modK : ''F'' → '''R''' by{{sfn|Weil|1995|p=4}} |
Given a local field ''F'', a "module function" on ''F'' 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 module of an element ''a'' of ''F'' is defined so as to measure the change in size of a set after multiplying it by ''a''. Specifically, define modK : ''F'' → '''R''' by{{sfn|Weil|1995|p=4}} |
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Using modK, one may then define an absolute value |.| on ''F'' that induces a [[Metric space|metric]] on ''F'' (via the standard d(''x'',''y'') = |''x''-''y''|), such that ''F'' is complete with respect to this metric, and the metric induces the given topology on ''F''. |
Using modK, one may then define an absolute value |.| on ''F'' that induces a [[Metric space|metric]] on ''F'' (via the standard d(''x'',''y'') = |''x''-''y''|), such that ''F'' is complete with respect to this metric, and the metric induces the given topology on ''F''. |
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==Basic features of non-Archimedean local fields== |
== Basic features of non-Archimedean local fields == |
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For a non-Archimedean local field ''F'' (with absolute value denoted by |·|), the following objects are important: |
For a non-Archimedean local field ''F'' (with absolute value denoted by |·|), the following objects are important: |
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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. |
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. |
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===Examples=== |
=== Examples === |
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#'''The ''p''-adic numbers''': the ring of integers of '''Q'''''p'' is the ring of ''p''-adic integers '''Z'''''p''. Its prime ideal is ''p'''''Z'''''p'' and its residue field is '''Z'''/''p'''''Z'''. Every non-zero element of '''Q'''p can be written as ''u'' ''p''''n'' where ''u'' is a unit in '''Z'''''p'' and ''n'' is an integer, with ''v''(''u'' ''p''n) = ''n'' for the normalized valuation. |
#'''The ''p''-adic numbers''': the ring of integers of '''Q'''''p'' is the ring of ''p''-adic integers '''Z'''''p''. Its prime ideal is ''p'''''Z'''''p'' and its residue field is '''Z'''/''p'''''Z'''. Every non-zero element of '''Q'''p can be written as ''u'' ''p''''n'' where ''u'' is a unit in '''Z'''''p'' and ''n'' is an integer, with ''v''(''u'' ''p''n) = ''n'' for the normalized valuation. |
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#The field '''C'''((''T'')) of formal Laurent series over the complex numbers is ''not'' a local field. Its residue field is '''C''' |
#The field '''C'''((''T'')) of formal Laurent series over the complex numbers is ''not'' a local field. Its residue field is '''C''' |
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===Higher unit groups=== |
=== Higher unit groups === |
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The '''''n''th higher unit group''' of a non-Archimedean local field ''F'' is |
The '''''n''th higher unit group''' of a non-Archimedean local field ''F'' is |
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for ''n'' ≥ 1.{{sfn|Neukirch|1999|p=122}} (Here "" means a non-canonical isomorphism.) |
for ''n'' ≥ 1.{{sfn|Neukirch|1999|p=122}} (Here "" means a non-canonical isomorphism.) |
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===Structure of the unit group=== |
=== 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 |
The multiplicative group of non-zero elements of a non-Archimedean local field ''F'' is isomorphic to |
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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 (geometry)|flag]] of subschemes of an ''n''-dimensional [[arithmetic scheme]]. |
From the geometric point of view, ''n''-dimensional local fields with last finite residue field are naturally associated to a complete [[flag (geometry)|flag]] of subschemes of an ''n''-dimensional [[arithmetic scheme]]. |
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==See also== |
== See also == |
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* [[Hensel's lemma]] |
* [[Hensel's lemma]] |
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* [[Ramification group]] |
* [[Ramification group]] |
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{{Reflist}} |
{{Reflist}} |
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==References== |
== References == |
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{{refbegin}} |
{{refbegin}} |
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*{{Citation|publisher=[[ |
*{{Citation |publisher=[[London Mathematical Society]] |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=2010 |orig-year=1967 |isbn=978-0-95027342-6}} |
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* {{Citation |
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* {{Citation |last1=Fesenko |first1=Ivan |author-link=Ivan Fesenko |last2=Vostokov |first2=Sergei |title=Local Fields and Their Extensions |publisher=[[American Mathematical Society]] |location=Providence |year=2002 |series=Translations of Mathematical Monographs |volume=121 |edition=2nd |orig-year=1993 |isbn=978-0-8218-3259-2 |mr=1915966}} |
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| last1=Fesenko |
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| first1=Ivan B. |
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| ⚫ | |||
| author-link=Ivan Fesenko |
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| last2=Vostokov |
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* {{Citation |last=Neukirch |first=Jürgen |author-link=Jürgen Neukirch |title=Algebraic Number Theory |publisher=Springer Berlin, Heidelberg |year=1999 |series=A Series of Comprehensive Studies in Mathematics |volume=322 |isbn=978-3-540-65399-8}} |
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| first2=Sergei V. |
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| title=Local fields and their extensions |
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* {{Citation |last=Serre |first=Jean-Pierre |author-link=Jean-Pierre Serre |title=[[Local Fields]] |publisher=Springer New York |year=1979 |series=Graduate Texts in Mathematics |volume=67 |isbn=978-0-387-90424-5}} |
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| publisher=[[American Mathematical Society]] |
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| location=Providence, RI |
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* {{Citation |last=Weil |first=André |author-link=André Weil |title=[[Basic Number Theory]] |year=1995 |orig-year=1974 |publisher=Springer Berlin, Heidelberg |series=Classics in Mathematics |isbn=978-3-540-58655-5 |edition=3rd}} |
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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|trans=true}} |
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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=Serre |
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| first=Jean-Pierre |
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| author-link=Jean-Pierre Serre |
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| title=Local Fields |
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| publisher=Springer-Verlag |
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| location=New York |
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| year=1979 |
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| series=Graduate Texts in Mathematics |
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| volume=67 |
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| edition=First |
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| isbn=0-387-90424-7 |
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}} |
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{{refend}} |
{{refend}} |
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==External links== |
== External links == |
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* {{springer|title=Local field|id=p/l060130}} |
* {{springer|title=Local field|id=p/l060130}} |
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