Solvable Groups and Solvability by Radicals

A finite group is solvable when it admits a chain from the identity to the whole group such that each subgroup is normal in the next and every successive quotient is abelian. Equivalently, repeatedly take the subgroup generated by commutators; the derived series must eventually reach the identity. A commutator measures failure to commute, so this process asks whether all noncommutativity can be removed in finitely many abelian layers.

Abelian groups are solvable, but solvable groups need not be abelian. The symmetric group S3 has the normal subgroup A3, giving quotients C3 and C2. Its noncommuting permutations therefore still admit a two-layer reduction. By contrast, A5 is nonabelian and simple, so its commutator subgroup is the whole group: it cannot descend through a nontrivial abelian quotient. Consequently S5, which contains A5 as its derived subgroup, is not solvable.

A radical tower adjoins elements whose positive integer powers already belong to the preceding field. Individual steps need not be normal: adjoining the real cube root of two omits its complex companions. After adjoining suitable roots of unity, however, automorphisms of a normal radical step act by multiplying a chosen root by a root of unity. Those actions produce abelian, and after refinement cyclic, layers. Taking normal closures and organizing the layers is essential to the theorem.

In characteristic zero, a polynomial is solvable by radicals if and only if the Galois group of its splitting field is solvable. The converse constructs radical extensions after supplying roots of unity and resolving the solvable group into cyclic layers. This is an existence criterion for containment in a radical extension. It does not require the original splitting field itself to be presented by a radical tower without enlarging it, and it does not promise a short or aesthetically simple formula.

The generic degree-n polynomial has Galois group Sn over its rational-function coefficient field. For n at least five this group is not solvable, proving that no universal radical formula handles arbitrary coefficients. This does not say that every polynomial of degree five lacks radical solutions. The polynomial x to the fifth minus two plainly has roots built from a fifth root of two and fifth roots of unity; its Galois group is a solvable semidirect product of orders five and four.

To establish impossibility for a particular polynomial, determine enough of its Galois group to prove nonsolvability. Irreducibility alone does not suffice. Conversely, numerical approximation remains available for nonsolvable groups: radical expressibility restricts permitted exact operations, not accurate decimal computation. Keeping these distinctions separate prevents an algebraic impossibility theorem from becoming a mistaken claim of general computational impossibility.