HilbertTheorem90 2002-01-07 12:44:12 2008-02-07 22:40:13 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $L/K$ be a finite Galois extension with Galois group $G = \operatorname{Gal}(L/K)$. 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.