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 ${\displaystyle (\cdot ,\cdot ):F^{*}\times F^{*}\rightarrow G}$, where G is an abelian group, written multiplicatively, such that

• ${\displaystyle (\cdot ,\cdot )}$ is bimultiplicative;
• if ${\displaystyle a+b=1}$ then ${\displaystyle (a,b)=1}$.

The symbols on F derive from a "universal" symbol, which may be regarded as taking values in ${\displaystyle F^{*}\otimes F^{*}/\langle a\otimes 1-a\rangle }$. By a theorem of Hideya Matsumoto, this group is ${\displaystyle K_{2}F}$ and is part of the Milnor K-theory for a field.

Properties

If (⋅,⋅) is a symbol then (assuming all terms are defined)

• ${\displaystyle (a,-a)=1}$;
• ${\displaystyle (b,a)=(a,b)^{-1}}$;
• ${\displaystyle (a,a)=(a,-1)}$ is an element of order 1 or 2;
• ${\displaystyle (a,b)=(a+b,-b/a)}$.

Examples

• The trivial symbol which is identically 1.
• The Hilbert symbol on F with values in {±1} defined by^[1][2]

${\displaystyle (a,b)={\begin{cases}1,&{\mbox{ if }}z^{2}=ax^{2}+by^{2}{\mbox{ has a non-zero solution }}(x,y,z)\in F^{3};\\-1,&{\mbox{ if not.}}\end{cases}}}$

• 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(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.^[3]

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.^[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.^[5]

See also

• Steinberg group (K-theory)