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 , 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 . By a theorem of Hideya Matsumoto, this group is and is part of the Milnor K-theory for a field.
If (â ,â ) is a symbol then (assuming all terms are defined)
If F is a topological field then a symbol c is weakly continuous if for each y in F<sup>âÂÂ</sup> the set of x in F<sup>âÂÂ</sup> such that c(x,y) = 1 is closed in F<sup>âÂÂ</sup>. 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<sub>2</sub>(F) is the direct sum of a cyclic group of order m and a divisible group K<sub>2</sub>(F)<sup>m</sup>. A symbol on F lifts to a homomorphism on K<sub>2</sub>(F) and is weakly continuous precisely when it annihilates the divisible component K<sub>2</sub>(F)<sup>m</sup>. It follows that every weakly continuous symbol factors through the norm residue symbol.