One interesting result in Galois theory is the Kronecker-Weber theorem: every finite abelian extension of $\mathbb{Q}$ (in $\mathbb{C}$) is inside some cyclotomic field.
The proof of this result uses some more machinary than the standard material for graduate course in Galois theory, namely, it uses heavy machinary from Algebraic Number Theory: splitting of primes, ramifications etc.
In the book on Galois theory by Escofier, the author points out one simple case in the form of simple exercises: quadratic extensions of $\mathbb{Q}$ are inside cyclotomic fields.
Question: I want to know if there are further more cases for finite abelian extensions of $\mathbb{Q}$, for which, it is easy to prove their embedding in cyclotomic fields, with no machinary from algebraic number theory (such as ramification etc.)
Best Answer
This is answered at this post, for the case that the extension has Galois group $\Bbb Z/p$, where $p$ is a regular prime. It naturally generalizes the quadratic case you have mentioned. The proof by David E Speyer doesn't need "heavy machinery" of ramification theory. It uses Stickelberger's Theorem, which is already contained in the elementary introduction to number theory by Rosen and Ireland.
Franz Lemmermeyer has written an excellent note about the general case using Kummer theory and Stickelberger's Theorem, see Kronecker-Weber via Stickelberger. This, however, uses of course ramification, but not so much machinery.