In mathematics a Steinberg symbol is a pairing function which generalises the

Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity ...

and plays a role in the

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...

of fields. It is named after mathematician

Robert Steinberg Robert Steinberg (May 25, 1922, Soroca, Bessarabia, Romania (present-day Moldova) – May 25, 2014) was a mathematician at the University of California, Los Angeles. He introduced the Steinberg representation, the Lang–Steinberg theorem, th ...

. 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 In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a field F) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebr ...

for a field.

Properties

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

(a, -a) = 1

; *

(b, a) = (a, b)^

; *

(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

on ''F'' with values in defined byMilnor (1971) p.94 :

(a,b)=\begin1,&\mboxz^2=ax^2+by^2\mbox(x,y,z)\in F^3;\\-1,&\mbox\end

*The

Contou-Carrère symbol In mathematics, the Contou-Carrère symbol 〈''a'',''b''〉 is a Steinberg symbol defined on pairs of invertible elements of the ring of Laurent power series In mathematics, the Laurent series of a complex function f(z) is a representation of ...

is a symbol for the ring of

Laurent power series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion ...

over an

Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...

Continuous symbols

If ''F'' is a

topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is ...

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 In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...

, then one may speak of a ''continuous symbol'', and when ''G'' is Hausdorff then a continuous symbol is weakly continuous.Milnor (1971) p.165 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.Milnor (1971) p.166 The characterisation of weakly continuous symbols on a non-Archimedean

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...

''F'' was obtained by Moore. The group K₂(''F'') is the direct sum of a

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...

of order ''m'' and a

divisible group In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive i ...

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 In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced ...

.Milnor (1971) p.175

References

* * * * {{cite journal , first=Robert , last=Steinberg , authorlink=Robert Steinberg , title=Générateurs, relations et revêtements de groupes algébriques , journal=Colloq. Théorie des Groupes Algébriques , location=Bruxelles , year=1962 , publisher=Gauthier-Villars , pages=113–127 , mr=0153677 , zbl=0272.20036 , language=French

External links

Steinberg symbol
at the Encyclopaedia of Mathematics K-theory

Properties

Examples

Continuous symbols

See also

References

External links