Hilbert symbol [SEP] named after
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" whose multiplicative group of non-zero elements is "K",
the quadratic Hilbert symbol is the function (–, –) from "K" × "K" to {−1,1} defined by
Quadratic Hilbert symbol Properties.
The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine
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
formula_1, where "G" is an abelian group, written multiplicatively, such that
- formula_2 is bimultiplicative;
- if formula_3 then formula_4.
The symbols on "F