Hilbert Symbol

In mathematics, given a local field K, such as the fields of reals or p-adic numbers, whose multiplicative group of non-zero elements is K×, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the formulation of local class field theory. As the name suggests, it was in some sense introduced by David Hilbert, although it would be anachronistic to say that of the local field formulation.

Explicitly, it is the function (–, –) from K× × K× to {−1,1} defined by