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
formula_1, where "G" is an abelian group, written multiplicatively, such that
The symbols on "F" derive from a "universal" symbol, which may be regarded as taking values in formula_5. By a theorem of Matsumoto, this group is formula_6 and is part of the Milnor K-theory for a field.
Properties.
If (⋅,⋅) is a symbol then (assuming all terms are defined)
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.