I was wondering what are the most efficient ways to find a cyclotomic field s.t. given $K/\mathbb{Q}$ we have $K\leq \mathbb{Q}(\zeta_n)$? For quadratic fields this is easy by just considering factors of $d$ for the adjoined square root $\sqrt{d}$. How about cubic extensions and higher?
I guess such a formula can't be solely based on the primes that ramify, because we could construct fields with discriminant of the form $p^aq^b$ for rather large $a$ and $b$, and this would require $n$ be large too.