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Steinberg symbol

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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.

Contents

For a field F we define a Steinberg symbol (or simply a symbol) to be a function ( ⋅ , ⋅ ) : F ∗ × F ∗ → G , where G is an abelian group, written multiplicatively, such that

  • ( ⋅ , ⋅ ) 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 ∗ ⊗ F ∗ / ⟨ a ⊗ 1 − a ⟩ . 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
    • ( a , b ) = { 1 ,  if  z 2 = a x 2 + b y 2  has a non-zero solution  ( x , y , z ) ∈ F 3 ; − 1 ,  if not.
  • 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.

    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. 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.

    References

    Steinberg symbol Wikipedia
    (Text) CC BY-SA


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