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 , where G is an abelian group, written multiplicatively, such that

  • is bimultiplicative;
  • if then .

The symbols on F derive from a "universal" symbol, which may be regarded as taking values in . By a theorem of Matsumoto, this group is and is part of the Milnor K-theory for a field.

Properties

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

  • ;
  • ;
  • is an element of order 1 or 2;
  • .

Examples

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

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

References

  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.
  2. Milnor (1971) p.94
  3. Milnor (1971) p.165
  4. Milnor (1971) p.166
  5. Milnor (1971) p.175
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