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.
, 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
If (⋅,⋅) 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∗ the set of x in F∗ such that c(x,y) = 1 is closed in F∗.
[4] 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.