If $mdc(m,n)=1$ then $\mathbb{Q}(\zeta_m,\zeta_n )=\mathbb{Q}(\zeta_{mn} )$,but what if the degree of the roots are divisors and multiples of each other?
I guess that $\mathbb{Q}(\zeta_4,\zeta_6, \zeta_{12} )=\mathbb{Q}(\zeta_{12})$
but what about $\mathbb{Q}(\zeta_{12},\zeta_8 )$?
Start with just adjoining the primitive 4th root of unity and 6th root of unity. If you multiply these two primitive roots together, what power is required to bring you back to $1$? What can you conclude?