Steinberg symbol

In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of 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]\displaystyle{ ( \cdot , \cdot ) : F^* \times F^* \rightarrow G }[/math], where G is an abelian group, written multiplicatively, such that

• [math]\displaystyle{ ( \cdot , \cdot ) }[/math] is bimultiplicative;
• if [math]\displaystyle{ a+b = 1 }[/math] then [math]\displaystyle{ (a,b) = 1 }[/math].

The symbols on F derive from a "universal" symbol, which may be regarded as taking values in [math]\displaystyle{ F^* \otimes F^* / \langle a \otimes 1-a \rangle }[/math]. By a theorem of Matsumoto, this group is [math]\displaystyle{ 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]\displaystyle{ (a, -a) = 1 }[/math];
• [math]\displaystyle{ (b, a) = (a, b)^{-1} }[/math];
• [math]\displaystyle{ (a, a) = (a, -1) }[/math] is an element of order 1 or 2;
• [math]\displaystyle{ (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

[math]\displaystyle{ (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^∗ the set of x in F^∗ such that c(x,y) = 1 is closed in F^∗. 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 then a continuous symbol is weakly continuous.

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. The characterisation of weakly continuous symbols on a non-Archimedean local field F was obtained by Moore. The group K₂(F) is the direct sum of a cyclic group of order m and a divisible group K₂(F)^m. A symbol on F lifts to a homomorphism on K₂(F) and is weakly continuous precisely when it annihilates the divisible component K₂(F)^m. It follows that every weakly continuous symbol factors through the norm residue symbol.

See also

• Steinberg group (K-theory)

References

• Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. 2. World Scientific. ISBN 9971-966-05-0. 
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. pp. 132–142. ISBN 0-8218-1095-2. 
• Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. 72. Princeton, NJ: Princeton University Press. 
• Steinberg, Robert (1962). "Générateurs, relations et revêtements de groupes algébriques" (in French). Colloq. Théorie des Groupes Algébriques (Bruxelles: Gauthier-Villars): 113–127. 

External links

• Steinberg symbol at the Encyclopaedia of Mathematics