Why require abelian when we define solvable group in Galois theory? In the definition on Wikipedia, it says

A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups $1 = G_0 < G_1 < ⋅⋅⋅ < G_k = G$ such that $G_{j−1}$ is normal in $G_j,$ and $G_j /G_{j−1}$ is an abelian group, for $j = 1, 2,\dots, k.$

We can regard this as two constraints:

*

*$G_{j−1}$ is normal in $G_j$

*$G_j /G_{j−1}$ is an abelian group

But why $G_j /G_{j−1}$ must be an abelian group? How could this relate to having a radical solution of a polynomial?
 A: I think this is a fair question. It's definitely not obvious what abelian has to do with radical!
When $G$ is a finite group, here is a different, equivalent (the equivalence only holds when $G$ is finite) definition of $G$ being solvable:

A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups $1=_0<G_1<\cdots<G_k=G$ such that $_{−1}$ is normal in $_$, and $_/_{−1}$ is a cyclic group, for $j = 1,\ldots, k$.

Now here's the point. By the fundamental theorem of Galois theory, the chain $1=_0<G_1<\cdots<G_k=G$ corresponds to a tower of extensions $L_0\supset\cdots \supset L_k$, and each $L_i/L_{i+1}$ is Galois with cyclic Galois group.
Now, there is an important theorem, due to Kummer:

Theorem: Let $L/K$ be a Galois extension with Galois group $C_n$. If $K$ contains an $n$-th root of unity, then there exists $\alpha \in K$ such that $L = K(\sqrt[n]\alpha)$.

In other words, assuming that $K$ contains enough roots of unity, cyclic extensions of $K$ are exactly the same as radical extensions. (The converse to Kummer's theorem is true and much easier!)
So here's how we solve a polynomial $f$ over $K$.

*

*Add enough roots of unity to $K$, depending on the degree of $K$. $K(\zeta_n)$ is always a radical extension.


*Compute the Galois group $G$ of $f$ over $K(\zeta_n)$.


*If $G$ is solvable, that means we can find a tower of cyclic Galois extensions above $K(\zeta_n)$. But, by Kummer's theorem, these extensions are radical extensions.
