In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''
× × ''K''
× to the group of ''n''th 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 in his
Zahlbericht
In mathematics, the ''Zahlbericht'' (number report) was a report on algebraic number theory by .
History
In 1893 the German mathematical society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski ...
, 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'' whose
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of non-zero elements is ''K''
×,
the quadratic Hilbert symbol is the
function (–, –) from ''K''
× × ''K''
× to defined by
:
Equivalently,
if and only if
is equal to the
norm of an element of the quadratic extension