A
group G is
solvable if there exist groups
G_0,
G_1, . . .,
G_
k such that:
- G_0 is the trivial group.
- G_k is G.
- G_i is a normal subgroup of G_{i+1}, for 0 <= i < k.
- G_{i+1} / G_i is abelian for 0 <= i < k.
An important theorem of
Galois theory is that a
polynomial can be
solved by radicals iff its
Galois group is solvable. One can construct a
quintic polynomial with Galois group
S_5---said group being unsolvable
[1]---so there can be no general
solution by radicals for polynomials of degree five or higher. That is, there is no
analogue of the
quadratic formula for polynomials of
degree >= 5.
[1]: S_5's only proper normal subgroups are e (the trivial group) and A_5. S_5 / e = S_5 is not abelian, so that doesn't work. S _5 / A_5 = Z_2 is abelian, but A_5, being simple and nonabelian, is not itself solvable. Hence S_5 cannot be solvable.