Galois-theoretic derivation of the quartic formula

Let x4+ax3+bx2+cx+d be a general polynomialPlanetmathPlanetmath with four roots r1,r2,r3,r4, so (x-r1)(x-r2)(x-r3)(x-r4)=x4+ax3+bx2+cx+d. The goal is to exhibit the field extension (r1,r2,r3,r4)/(a,b,c,d) as a radical extension, thereby expressing r1,r2,r3,r4 in terms of a,b,c,d by radicalsPlanetmathPlanetmathPlanetmathPlanetmath.

Write N for (r1,r2,r3,r4) and F for (a,b,c,d). The Galois groupMathworldPlanetmath Gal(N/F) is the symmetric groupMathworldPlanetmathPlanetmath S4, the permutation groupMathworldPlanetmath on the four elements {r1,r2,r3,r4}, which has a composition seriesMathworldPlanetmathPlanetmath



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 groupsMathworldPlanetmath: