# Galois criterion for solvability of a polynomial by radicals

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 $\deg(f)$.

Title Galois criterion for solvability of a polynomial by radicals GaloisCriterionForSolvabilityOfAPolynomialByRadicals 2013-03-22 12:08:58 2013-03-22 12:08:58 djao (24) djao (24) 7 djao (24) Theorem msc 11R32