Galois-theoretic derivation of the quartic formula
Let x4+ax3+bx2+cx+d be a general polynomial 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
ℂ(r1,r2,r3,r4)/ℂ(a,b,c,d) as a radical extension, thereby
expressing r1,r2,r3,r4 in terms of a,b,c,d by radicals
.
Write N for ℂ(r1,r2,r3,r4) and F for ℂ(a,b,c,d). The
Galois group Gal(N/F) is the symmetric group
S4, the permutation group
on the four elements {r1,r2,r3,r4}, which has a composition series
1⊲ |
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: