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

is bimultiplicative;
if then.

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.

Properties

If is a symbol then

;
;
is an element of order 1 or 2;
.

Examples

The trivial symbol which is identically 1.
The Hilbert symbol on F with values in defined by
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 = 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 K₂^m. A symbol on F lifts to a homomorphism on K₂ and is weakly continuous precisely when it annihilates the divisible component K₂^m. It follows that every weakly continuous symbol factors through the norm residue symbol.