abstract algebra - $mathbb Q(zeta_m)capmathbb Q(zeta_n)=mathbb Q(zeta_d)$

Prove that $\mathbb Q(\zeta_m)\cap\mathbb Q(\zeta_n)=\mathbb Q(\zeta_d)$ where $d=\gcd(m,n)$.

I want to solve this problem without Galois theory.

I know only about field extension. For example, algebraic extension, cyclotomic extension, splitting field and algebraic closure.

Can I solve it without Galois theory?