Galoistheoretic derivation of the quartic formula
Let ${x}^{4}+a{x}^{3}+b{x}^{2}+cx+d$ be a general polynomial^{} with four roots ${r}_{1},{r}_{2},{r}_{3},{r}_{4}$, so $(x{r}_{1})(x{r}_{2})(x{r}_{3})(x{r}_{4})={x}^{4}+a{x}^{3}+b{x}^{2}+cx+d$. The goal is to exhibit the field extension $\u2102({r}_{1},{r}_{2},{r}_{3},{r}_{4})/\u2102(a,b,c,d)$ 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 $\u2102({r}_{1},{r}_{2},{r}_{3},{r}_{4})$ and $F$ for $\u2102(a,b,c,d)$. The Galois group^{} $\mathrm{Gal}(N/F)$ 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^{}
$$1\u22b2\mathbb{Z}/2\u22b2{V}_{4}\u22b2{A}_{4}\u22b2{S}_{4},$$ 
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 fourgroup^{}.

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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^{}: