Main Page

2014-08-18T19:54:26Z

NigelCarbajal:

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.

==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 [[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>
:<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>

==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 | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | pages=132–142 }}
* {{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]]

{{algebra-stub}}

formulasearchengine - User contributions [en]

Main Page