# Ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$

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?

• $\Bbb Q(\zeta_n,\zeta_m)=\Bbb Q(\zeta_{{\rm lcm}(n,m)})$ and ${\cal O}_{\Bbb Q(\zeta_n)}=\Bbb Z[\zeta_n]$ for all $n$ (proving this second fact is much more involved, and is easiest for prime $n$).
– anon
Feb 14, 2014 at 15:18

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 $\mathbb{Q}(\zeta_p,\zeta_q)=\mathbb{Q}(\zeta_d)$.