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

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

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

Properties

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

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

Examples

• The Hilbert symbol on F with values in {±1} defined by

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

• The Contou-Carrère symbol is a symbol for the ring of Laurent power series over an Artinian ring.

See also

• Steinberg group (K-theory)

References

• 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.
• 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.{{cite journal}}: CS1 maint: MR format (link)

External links

• Steinberg symbol at the Encyclopaedia of Mathematics