At the end of the fundamental theorem of Galois theory, and after some intermediate moments of clarity realizing, for example, that the subfield lattice is built on fixed elements by different automorphisms, and that the corresponding subgroups are permutations of roots, there comes the idea that the quintic doesn't have necessarily radical solutions because we can find polynomials with a subgroup lattice that is not "solvable."
Unfortunately, and despite the suggestive name of "solvable," it feels like hitting an unmotivated new definition involving a chain of normal subgroups and abelian quotient groups.
What is the non-axiomatic intuition of the solvability of subgroups to see their need to have radical solutions?
Best Answer
There's no real intuition behind calling solvable groups solvable other than the fact that a solvable Galois group is equivalent to solvability by radicals. But in a typical lecture, the definition comes first, and then the proof that solvability of the Galois group and solvability of the equation by radicals are equivalent. Historically, this was the other way around: People were looking for conditions under which a polynomial equation is solvable by radicals. When they actually found a neccessary and sufficient condition, they called it solvability, since that's what it's really used for: determining wether polynomials equations are solvable (by radicals) or not.
Solving a polynomial equation by radicals is essentially nothing else than taking the field we're working with, throwing a radical into the mix, then adding another radical, then another, and so on, until the solution of the polynomial equation is included in the field. If this is possible, then the equation is solvable by radicals. And the nice thing is, we can throw in additional, possibly unneeded radicals to make working with the corresponding field extensions easier. It would be optimal if we could make the extensions Galois! To start, we can throw in as many roots of unity as we like. So we do that. Luckily, adding primitive $n$-th roots of unity to $\mathbb Q$ results in a Galois extension with Galois group $\mathbb Z_n^\times$ (the unit group of $\mathbb Z_n$), which is Abelian. Now comes the important part: If our field already contains suitable roots of unity, then adding a root of any other element will also result in a Galois extension, this time with cyclic Galois group. This means that as we add all the roots we need to solve our polynomial equation, we always get a Galois extension with Abelian Galois group (cyclic groups are also Abelian). Now consider the finished extension and its Galois group. All the Abelian groups we considered beforehand are the factor groups in the chain of normal subgroups used to define solvable groups!
So, if a polynomial equation is solvable by radicals, then there is a Galois extension in which the polynomial splits and which has a chain of normal subgroups with Abelian factors. And we can also show that if a group has this property, then its subgroups have this property as well. And the Galois group of the splitting field of our polynomial is a subgroup of the one we just constructed. So if a polynomial equation is solvable, then the Galois group of its splitting field has the property that there exists a chain of normal subgroups with Abelian factors. This is big, since it gives us a neccessary condition for solvability by radicals.
Even better! It can be shown that this condition is not only neccessary. It is also sufficient. So this property of the Galois group completely determines wether the underlying equation is solvable by radicals. Such a cool property deserves a name. And since it completely determines solvability of an equation, why not call the property solvability?