In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert (1897, sections 64, 131, 1998, English translation) in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields.
Quadratic Hilbert symbol Over a local field K {\displaystyle K} with multiplicative group of non-zero elements K × {\displaystyle K^{\times }} , the quadratic Hilbert symbol is the function K × × K × → { ± 1 } {\displaystyle K^{\times }\times K^{\times }\to \{\pm 1\}} defined by
( a , b ) = { + 1 , if z 2 = a x 2 + b y 2 has a non-zero solution ( x , y , z ) ∈ K 3 ; − 1 , otherwise. {\displaystyle (a,b)={\begin{cases}+1,&{\mbox{ if }}z^{2}=ax^{2}+by^{2}{\mbox{ has a non-zero solution }}(x,y,z)\in K^{3};\\-1,&{\mbox{ otherwise.}}\end{cases}}}
Equivalently, ( a , b ) = 1 {\displaystyle (a,b)=1} if and only if b {\displaystyle b} is equal to the norm of an element of the quadratic extension K [ a ] {\displaystyle K[{\sqrt {a}}]} .
Properties The following three properties follow directly from the definition, by choosing suitable solutions of the Diophantine equation above:
If a {\displaystyle a} is a square, then ( a , b ) = 1 {\displaystyle (a,b)=1} for all b {\displaystyle b} . For all a , b {\displaystyle a,b} in K × {\displaystyle K^{\times }} , ( a , b ) = ( b , a ) {\displaystyle (a,b)=(b,a)} . For any a {\displaystyle a} in K × {\displaystyle K^{\times }} such that a − 1 {\displaystyle a-1} is also in K × {\displaystyle K^{\times }} , we have ( a , 1 − a ) = 1 {\displaystyle (a,1-a)=1} . The (bi)multiplicativity, i.e.,
( a , b 1 b 2 ) = ( a , b 1 ) ⋅ ( a , b 2 ) {\displaystyle (a,b_{1}b_{2})=(a,b_{1})\cdot (a,b_{2})}
for any a , b 1 {\displaystyle a,b_{1}} and b 2 {\displaystyle b_{2}} in K × {\displaystyle K^{\times }} is, however, more difficult to prove, and requires the development of local class field theory. The third property shows that the Hilbert symbol is an example of a Steinberg symbol and thus factors over the second Milnor K-group K 2 M ( K ) {\displaystyle K_{2}^{M}(K)} , which is by definition
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