Hilbert symbol

Summary

The Hilbert symbol $(a, b)_K$ of two nonzero elements of a local field $K$ of characteristic other than $2$ is $1$ or $-1$. It is $1$ exactly when the equation $z^2 = a x^2 + b y^2$ has a solution other than $(0, 0, 0)$. It is symmetric and bimultiplicative, and it is a basic tool for quadratic forms over local fields. Over $\Q$, the product of all local Hilbert symbols is $1$, a statement equivalent to quadratic reciprocity.