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:

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$A_{4}$ is the alternating group in $S_{4}$ , consisting of the even permutations.
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$V_{4}=\{1,(12)(34),(13)(24),(14)(23)\}$ is the Klein four-group.
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$\mathbb{Z}/2$ 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: