As a math undegraduate, I was doing my homework of fields and Galois theory course and observed that many questions asked us to calculate the group $Gal(\mathbb{Q}(\xi_n)/\mathbb{Q})$, where $\xi_n$ is a primitive $n$th root of the unit (so, to say if it is isomorphic to $D_4$, to $C_2\times C_2\times C_3$, ... Or whatsoever...)
So, I imagine that such a thing so cannonical must be already classified, maybe not for every $n$, but for quite many of them. And I wondered if there would be any resource with a full classification of this kind of Galois groups. Or maybe there is a simple method common to all of them.
I have the feeling that this will have to do with cyclotomic extensions, but we have not yet covered those on my course.
