Galois-theoretic derivation of the quartic formula
Let be a general polynomial with four roots, so . The goal is to exhibit the field extension as a radical extension, therebyexpressing in terms of by radicals
.
Write for and for . TheGalois group is the symmetric group
, the permutation group
on the four elements , which has a composition series
where:
- •
is the alternating group
in , consisting of the evenpermutations
.
- •
is the Klein four-group
.
- •
is the two–element subgroup
of .
Under the Galois correspondence, each of these subgroups correspondsto an intermediate field of the extension . We denote these fixedfields by (in increasing order) , , and .
We thus have a tower of field extensions, and correspondingautomorphism groups: