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, thereby
expressing in terms of by radicals
.
Write for and for . The
Galois 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 even permutations
.
-
•
is the Klein four-group
.
-
•
is the two–element subgroup
of .
Under the Galois correspondence, each of these subgroups corresponds to an intermediate field of the extension . We denote these fixed fields by (in increasing order) , , and .
We thus have a tower of field extensions, and corresponding
automorphism groups: