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

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

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

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

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

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

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

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]

• Steinberg group (K-theory)

• Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. Vol. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. pp. 132–142. ISBN 0-8218-1095-2. Zbl 1068.11023.
• Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. Vol. 72. Princeton, NJ: Princeton University Press. MR 0349811. Zbl 0237.18005.
• Steinberg, Robert (1962). "Générateurs, relations et revêtements de groupes algébriques". Colloq. Théorie des Groupes Algébriques (in French). Bruxelles: Gauthier-Villars: 113–127. MR 0153677. Zbl 0272.20036.

• Steinberg symbol at the Encyclopaedia of Mathematics

• K-theory

• CS1 French-language sources (fr)