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 $$( \cdot, \cdot ) : F^* \times F^* \rightarrow G$$, where G is an abelian group, written multiplicatively, such that
 * $$( \cdot, \cdot ) $$ is bimultiplicative;
 * if $$a+b = 1$$ then $$(a,b) = 1$$.

The symbols on F derive from a "universal" symbol, which may be regarded as taking values in $$F^* \otimes F^* / \langle a \otimes 1-a \rangle$$. By a theorem of Matsumoto, this group is $$K_2 F$$ and is part of the Milnor K-theory for a field.

Properties
If (⋅,⋅) is a symbol then (assuming all terms are defined)
 * $$ (a, -a) = 1 $$;
 * $$ (b, a) = (a, b)^{-1} $$;
 * $$ (a, a) = (a, -1) $$ is an element of order 1 or 2;
 * $$ (a, b) = (a+b, -b/a) $$.

Examples

 * The trivial symbol which is identically 1.
 * The Hilbert symbol on F with values in {±1} defined by
 * $$(a,b)=\begin{cases}1,&\mbox{ if }z^2=ax^2+by^2\mbox{ has a non-zero solution }(x,y,z)\in F^3;\\-1,&\mbox{ if not.}\end{cases}$$


 * 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.