Steinberg symbol
# 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 $K_2(F)$ is the direct sum of a cyclic group of order m and a divisible group $K_2(F)^m.$ A symbol on F lifts to a homomorphism on $K_2(F)$ and is weakly continuous precisely when it annihilates the divisible component $K_2(F)^m.$ It follows that every weakly continuous symbol factors through the norm residue symbol.

