# Hilbert symbol

Let $K$ be any local field^{}. For any two nonzero elements $a,b\in {K}^{\times}$, we define:

$$(a,b):=\{\begin{array}{cc}+1\hfill & \text{if}{z}^{2}=a{x}^{2}+b{y}^{2}\text{has a nonzero solution}(x,y,z)\ne (0,0,0)\text{in}{K}^{3}\text{,}\hfill \\ -1\hfill & \text{otherwise.}\hfill \end{array}$$ |

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

Title | Hilbert symbol |
---|---|

Canonical name | HilbertSymbol |

Date of creation | 2013-03-22 12:50:48 |

Last modified on | 2013-03-22 12:50:48 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 11S31 |

Classification | msc 11S80 |