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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

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where:

  • A4 is the alternating groupMathworldPlanetmath in S4, consisting of the even permutationsMathworldPlanetmath.

  • V4={1,(12)(34),(13)(24),(14)(23)} is the Klein four-groupMathworldPlanetmath.

  • /2 is the two–element subgroupMathworldPlanetmathPlanetmath {1,(12)(34)} of V4.

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: