Hilbert symbol

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 $K$ .