Galois group GaloisGroup 2002-01-05 02:20:53 2004-07-28 16:19:46 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} The \emph{Galois group} $\operatorname{Gal}(K/F)$ of a field extension $K/F$ is the group of all field automorphisms $\sigma\colon K \to K$ of $K$ which fix $F$ (i.e., $\sigma(x) = x$ for all $x \in F$). The group operation is given by composition: for two automorphisms $\sigma_1, \sigma_2 \in \operatorname{Gal}(K/F)$, given by $\sigma_1\colon K \to K$ and $\sigma_2\colon K \to K$, the product $\sigma_1 \cdot \sigma_2 \in \operatorname{Gal}(K/F)$ is the composite of the two maps $\sigma_1 \circ \sigma_2\colon K \to K$. The \emph{Galois group} of a polynomial $f(x) \in F[x]$ is defined to be the Galois group of the splitting field of $f(x)$ over $F$.