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

(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]

References

Serre, Jean-Pierre (1996). A Course in Arithmetic. Graduate Texts in Mathematics. 7. Berlin, New York: Springer-Verlag. ISBN 978-3-540-90040-5.
Milnor (1971) p.94
Milnor (1971) p.165
Milnor (1971) p.166
Milnor (1971) p.175

Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. 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. 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. 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.

External links

Steinberg symbol at the Encyclopaedia of Mathematics

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[1] Serre, Jean-Pierre (1996). A Course in Arithmetic. Graduate Texts in Mathematics. 7. Berlin, New York: Springer-Verlag. ISBN 978-3-540-90040-5.

[Mil94-2] Milnor (1971) p.94

[Mil165-3] Milnor (1971) p.165

[Mil166-4] Milnor (1971) p.166

[Mil175-5] Milnor (1971) p.175