Hilbert reciprocity law (${\prod }_v(a,b{)}_v=1$)

For nonzero $a,b$ in a number field, the product of their quadratic Hilbert symbols over all places is one. All but finitely many symbols are one. Over $\mathbb{Q}$, this is equivalent to quadratic reciprocity with its supplementary laws. It supplies the compatibility relation between the local invariants of a global quadratic form.