Galois-theoretic derivation of the quartic formula

Let $x^{4}+ax^{3}+bx^{2}+cx+d$ be a general polynomial with four roots $r_{1},r_{2},r_{3},r_{4}$, so $(x-r_{1})(x-r_{2})(x-r_{3})(x-r_{4})=x^{4}+ax^{3}+bx^{2}+cx+d$. The goal is to exhibit the field extension $\mathbb{C}(r_{1},r_{2},r_{3},r_{4})/\mathbb{C}(a,b,c,d)$ as a radical extension, thereby expressing $r_{1},r_{2},r_{3},r_{4}$ in terms of $a,b,c,d$ by radicals.

Write $N$ for $\mathbb{C}(r_{1},r_{2},r_{3},r_{4})$ and $F$ for $\mathbb{C}(a,b,c,d)$. The Galois group $\operatorname{Gal}(N/F)$ is the symmetric group $S_{4}$, the permutation group on the four elements $\{r_{1},r_{2},r_{3},r_{4}\}$, which has a composition series

 $1\lhd\mathbb{Z}/2\lhd V_{4}\lhd A_{4}\lhd S_{4},$

where:

• $A_{4}$ is the alternating group in $S_{4}$, consisting of the even permutations.

• $V_{4}=\{1,(12)(34),(13)(24),(14)(23)\}$ is the Klein four-group.

• $\mathbb{Z}/2$ is the two–element subgroup $\{1,(12)(34)\}$ of $V_{4}$.

Under the Galois correspondence, each of these subgroups corresponds to an intermediate field of the extension $N/F$. We denote these fixed fields by (in increasing order) $K$, $L$, and $M$.

We thus have a tower of field extensions, and corresponding automorphism groups: