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 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 Matsumoto, this group is and is part of the Milnor K-theory for a field.

Properties

If is a symbol then
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 = 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 is the direct sum of a cyclic group of order m and a divisible group K2m. A symbol on F lifts to a homomorphism on K2 and is weakly continuous precisely when it annihilates the divisible component K2m. It follows that every weakly continuous symbol factors through the norm residue symbol.