en>Monkbot: Task 2: Fix CS1 deprecated coauthor parameter errors

2014-07-26T17:47:17Z

Task 2: Fix CS1 deprecated coauthor parameter errors

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	In mathematics a '''Steinberg symbol''' is a pairing function which generalises the [[Hilbert symbol]] and plays a role in the [[algebraic K-theory]] of [[field (mathematics)\|fields]]. It is named after mathematician [[Robert Steinberg]].

	~~For a field ''F'' we define a ''Steinberg symbol'' (or simply a ''symbol'') to be a function~~
	~~<math>( \cdot , \cdot ) : F^* \times F^* \rightarrow G</math>, where ''G'' is an abelian group, written multiplicatively, such that~~
	* <math>( \cdot , \cdot ) </math> is bimultiplicative;
	* if <math>a+b = 1</math> then <math>(a,b) = 1</math>.

	~~The symbols~~ on ~~''F'' derive from~~ a ~~"universal" symbol~~, ~~which may be regarded as taking values in <math>F^* \otimes F^* / \langle a \otimes~~ 1~~-a \rangle</math>~~. ~~By a theorem~~ of ~~Matsumoto,~~ this ~~group is <math>K_2 F</math> and is part of the [[Milnor K-theory]]~~ for a ~~field~~.		Rupert Brown regularly contributes articles on dog beds. Pack a timer that you can attach to your lights that will automatically shut them off after a specific time. Any ordinary individual would probably hop in their automobile, drive to the place they bought it, and get 1. Pause at the end of this paragraph – just for a few seconds – to recall a very happy, joyful, engaging moment from your life. 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	~~==Properties==~~
	If ~~(⋅,⋅) is a symbol then (assuming all terms~~ are ~~defined)~~
	* <math> (a, ~~-a) = 1 </math>;~~
	* <math> (b, ~~a) = (a~~, ~~b)^{-1} </math>;~~
	* <math> (a, ~~a) = (a, -1) </math> is an element of order 1 or 2;~~
	* <math> (a, b) = (a+b, -b/~~a) <~~/~~math>~~.

	~~==Examples==~~
	* The trivial symbol which is identically 1.

	* The [[Hilbert symbol]] on ''F'' with values in {±1} defined by<ref>{{cite book \| last=~~Serre \| first~~=~~Jean-Pierre \| authorlink~~=~~Jean-Pierre Serre \| title~~=~~A Course in Arithmetic \| publisher~~=~~[[Springer-Verlag~~]~~] \| location=Berlin, New York \| series=~~[~~[Graduate Texts in Mathematics]] \| volume=7 \| isbn=978-3-540-90040-5 \| year=1996 }}<~~/~~ref><ref name=Mil94>Milnor (1971) p~~.94<~~/ref~~>
	:<~~math~~>~~(a,b)=\begin{cases}1,&\mbox{ if }z^2=ax^2+by^2\mbox{ has a non-zero solution }(x,y,z)\~~in ~~F^3;\\-1~~,~~&\mbox{ if not~~.~~}\end{cases}</math>~~

	*The ~~[[Contou-Carrère symbol]]~~ is a ~~symbol for~~ the ~~ring~~ of ~~[[Laurent power series]] over an [[Artinian ring]]~~.

	~~==Continuous symbols==~~
	~~If ''F''~~ is a [~~[topological field~~]~~] then a symbol ''c'' is ''weakly continuous'' if for~~ each ~~''y'' in ''F''<sup>&lowast;</sup>~~ the ~~set~~ of ~~''x'' in ''F''~~<~~sup~~>~~&lowast;~~<~~/sup~~> ~~such~~ that '~~'c''(''x''~~,~~''y'') = 1~~ is ~~[[closed set\|closed]]~~ in ''~~F''~~<~~sup~~>~~&lowast;~~<~~/sup~~>. ~~This makes no reference to~~ a ~~topology~~ on the ~~codomain ''G''~~. ~~If ''G'' is a [[topological group]]~~, ~~then one may speak of a ''continuous symbol''~~, and ~~when ''G'' is [[Hausdorff space\|Hausdorff]] then~~ a ~~continuous symbol is weakly continuous~~.~~<ref name=Mil165>Milnor (1971) p~~.~~165</ref>~~

	~~The~~ only ~~weakly continuous symbols on '''R''' are~~ the ~~trivial symbol~~ and the ~~Hilbert symbol:~~ the ~~only weakly continuous symbol on '''C''' is the trivial symbol~~.<~~ref name=Mil166~~>~~Milnor (1971) p.166~~<~~/ref~~> ~~The characterisation~~ of ~~weakly continuous symbols~~ on a ~~non-Archimedean [[local field]] ''F'' was obtained by Moore~~. ~~The group K<sub>2</sub>(''F'') is~~ the ~~direct sum~~ of ~~a [[cyclic group]]~~ of ~~order ''m'' and~~ a ~~[[divisible group]] K<sub>2</sub>(''F'')<sup>''m''</sup>~~. ~~A symbol on ''F'' lifts~~ to a ~~homomorphism on K<sub>2</sub>(''F'')~~ and ~~is weakly continuous precisely when it annihilates~~ the ~~divisible component K<sub>2</sub>(''F'')<sup>''m''</sup>~~. It ~~follows~~ that ~~every weakly continuous symbol factors through the [[norm residue symbol]]~~.<~~ref name=Mil175~~>~~Milnor (1971) p.175~~<~~/ref~~>

	~~==See also==~~
	* [[Steinberg group (K-theory)]]

	~~==References==~~
	~~{{reflist}}~~
	* {{cite book \| first1=P.E. ~~\| last1=Conner \| first2=R. \| last2=Perlis \| title=A Survey~~ of ~~Trace Forms of Algebraic Number Fields \| series=Series in Pure Mathematics \| volume=2 \| publisher=World Scientific \| year=1984 \| isbn=9971-966-05-0 \| zbl=0551~~.~~10017 }}~~
	* {{cite book \| title=Introduction to ~~Quadratic Forms~~ over ~~Fields \| volume=67 \| series=Graduate Studies~~ in ~~Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| pages=132–142 \| zbl=1068~~.~~11023 }}~~
	* {{cite book \| last1=Milnor \| first1=John Willard \| author1-link= John Milnor \| title=Introduction to ~~algebraic K-theory \| publisher=[[Princeton University Press]] \| location=Princeton~~, ~~NJ \| mr=0349811 \| year=1971 \| zbl=0237.18005 \| series=Annals~~ of ~~Mathematics Studies \| volume=72 }}~~
	* {{cite journal \| first=Robert \| last=Steinberg \| authorlink=Robert Steinberg \| title=Générateurs, relations et revêtements de groupes algébriques \| journal=Colloq. Théorie des Groupes Algébriques \| location=Bruxelles \| year=1962 \| publisher=Gauthier-Villars \| pages=113–127 \| mr=MR0153677 \| zbl=0272.~~20036 \| language=French }}~~

	~~==External links==~~
	*[http://eom.springer.de/S/s130540.htm Steinberg symbol] at the [[Encyclopaedia of Mathematics]]

	~~[[Category:K-theory]]~~

en>Christian75: /* Comparison with other techniques */Clean up, typos fixed: et al → et al. using AWB

2013-04-09T10:57:01Z

Comparison with other techniques: Clean up, typos fixed: et al → et al. using AWB

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en>Bibcode Bot: Adding 0 arxiv eprint(s), 3 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2012-07-31T04:21:24Z

Adding 0 arxiv eprint(s), 3 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

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Reaction field method - Revision history

en>Monkbot: Task 2: Fix CS1 deprecated coauthor parameter errors

en>Christian75: /* Comparison with other techniques */Clean up, typos fixed: et al → et al. using AWB

en>Bibcode Bot: Adding 0 arxiv eprint(s), 3 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot