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 %$(x,y,z)\eq(0,0,0)$ in $K^{3}$,}\\\\-1&\\text{ otherwise.}\\end{cases}

The number $(a,b)$ is called the Hilbert symbol of $a$ and $b$ in $K$ .

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