Let $p,q$ be distinct odd primes, and $\omega_p,\omega_q$ primitive $p$-th, $q$-th roots of unity. What is the ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$?
The numbers in the ring $\mathbb{Q}(\omega_p,\omega_q)$ are of the form $\sum_{0\leq i<p,0\leq j<q}a_{ij}\omega_p^i\omega_q^j$ with $a_{ij}\in \mathbb{Q}$. To be in the ring of integers means the minimal polynomial is monic.
How can we compute the minimum polynomial here?
The first question is answered in Adjoining two primitive n-th roots, and that the ring of integers of $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}(\zeta_n)$ can be found in (almost) any lecture note on algebraic number theory, i.e., for example it is proved in Milne's lecture notes, chapter $6$. For the minimal polynomial, it is the cyclotomic polynomial for $d=\gcd(p,q)$ because $\mathbb{Q}(\zeta_p,\zeta_q)=\mathbb{Q}(\zeta_d)$.