Let $K$ be any local field. For any two nonzero elements $a,b \in K^\times$ we define: $$ (a,b) := \begin{cases} +1 & \text{ if $z^2 = ax^2 + by^2$ has a nonzero solution $(x,y,z) \neq (0,0,0)$ in $K^3$,} \\ -1 & \text{ otherwise.} \end{cases} $$ The number $(a,b)$ is called the Hilbert symbol of $a$ and $b$ in $K$