# 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 $K$.

Title Hilbert symbol HilbertSymbol 2013-03-22 12:50:48 2013-03-22 12:50:48 djao (24) djao (24) 5 djao (24) Definition msc 11S31 msc 11S80