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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.
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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]]

Latest revision as of 19:47, 26 July 2014


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