GaloisCriterionForSolvabilityOfAPolynomialByRadicals 2002-01-05 04:46:17 2005-10-11 13:01:22 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $f \in F[x]$ be a polynomial over a field $F$, and let $K$ be its splitting field. If $K$ is a radical extension of $F$, then the Galois group $\operatorname{Gal}(K/F)$ is a solvable group. Conversely, if the Galois group $\operatorname{Gal}(K/F)$ is a solvable group, then $K$ is a radical extension of $F$ provided that the characteristic of $K$ is either $0$ or greater than $\deg(f)$.