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 {\displaystyle F} we define a Steinberg symbol (or simply a symbol) to be a function
( ⋅ , ⋅ ) : F ∗ × F ∗ → G {\displaystyle (\cdot ,\cdot ):F^{*}\times F^{*}\rightarrow G} , where G {\displaystyle G} is an abelian group, written multiplicatively, such that for all a , b , c ∈ F ∗ {\displaystyle a,b,c\in F^{*}} ,
( a b , c ) = ( a , c ) ( b , c ) {\displaystyle (ab,c)=(a,c)(b,c)} and ( a , b c ) = ( a , b ) ( a , c ) {\displaystyle (a,bc)=(a,b)(a,c)} , and if a + b = 1 {\displaystyle a+b=1} then ( a , b ) = 1 {\displaystyle (a,b)=1} . The first condition is sometimes referred to as ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} being bimultiplicative, while the second condition is known as the Steinberg property. The symbols on F {\displaystyle F} derive from a "universal" symbol, which may be regarded as taking values in F ∗ ⊗ F ∗ / ⟨ a ⊗ 1 − a ⟩ {\displaystyle F^{*}\otimes F^{*}/\langle a\otimes 1-a\rangle } . By a theorem of Hideya Matsumoto, this group is K 2 F {\displaystyle K_{2}F} and is part of the Milnor K-theory for a field.
Properties If ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} is a symbol on F {\displaystyle F} , then for all a , b ∈ F ∗ {\displaystyle a,b\in F^{*}} we have
( a , − a ) = 1 {\displaystyle (a,-a)=1} ;
( b , a ) = ( a , b ) − 1 {\displaystyle (b,a)=(a,b)^{-1}} ;
( a , a ) = ( a , − 1 ) {\displaystyle (a,a)=(a,-1)} is an element of order 1 or 2;
( a , b ) = ( a + b , − b / a ) {\displaystyle (a,b)=(a+b,-b/a)} .
Examples The trivial symbol which is identically 1. The Hilbert symbol on F {\displaystyle 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. {\displaystyle (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.
… excerpt ends here. Continue reading the full article.
