# Hilbert Theorem 90

Let $L/K$ be a finite Galois extension^{} with Galois group^{} $G=\mathrm{Gal}(L/K)$. The modern formulation of Hilbert’s Theorem 90 states that the first Galois cohomology group ${H}^{1}(G,{L}^{*})$ is 0.

The original statement of Hilbert’s Theorem 90 differs somewhat from the modern formulation given above, and is nowadays regarded as a corollary of the above fact. In its original form, Hilbert’s Theorem 90 says that if $G$ is cyclic with generator^{} $\sigma $, then an element $x\in L$ has norm 1 if and only if

$$x=y/\sigma (y)$$ |

for some $y\in L$. Note that elements of the form $y/\sigma (y)$ are obviously contained within the kernel of the norm map; it is the converse that forms the content of the theorem.

Title | Hilbert Theorem 90 |
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Canonical name | HilbertTheorem90 |

Date of creation | 2013-03-22 12:10:41 |

Last modified on | 2013-03-22 12:10:41 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 7 |

Author | djao (24) |

Entry type | Theorem |

Classification | msc 11R32 |

Classification | msc 11S25 |

Classification | msc 11R34 |

Synonym | Hilbert’s Theorem 90 |

Synonym | Satz 90 |