Motivation: Let $F$ be a field of characteristic zero and let $f\in F[X]$. Then the equation $f(X)=0$ is solvable in radicals if and only if $\text{Gal}(f)$ is a solvable group (Theorem of Galois). On the other hand, every finite abelian group is the Galois group of a subextension of a cyclotomic extension (Kronecker-Weber Theorem).
It seems that there is not a lot of talk about "the middle ground", i.e. nilpotent groups, in Galois theory, so I'm wondering about the following:
What can be said about the equation $f(X)=0$ if $\text{Gal}(f)$ is nilpotent?
Is every finite nilpotent group a Galois group of some field extension of the rational numbers? If so, is there an easily comprehensible class of field extensions (like subextensions of cyclotomic extensions) having all finite nilpotent groups as Galois groups?
Any references would be helpful!