Solution
= Solution
Because $\operatorname{lcm}(9,15)=45$ and suitable <integral> powers of $\zeta_9$ and $\zeta_{15}$ generate a primitive forty-fifth root,
$$
L_1L_2=\mathbb Q(\zeta_{45}).
$$
The degree formula gives
$$
[L_1\cap L_2:\mathbb Q]
=\frac{\varphi(9)\varphi(15)}{\varphi(45)}
=\frac{6\cdot8}{24}=2.
$$
Both fields contain $\mathbb Q(\zeta_3)$, which already has degree two, so
$$
L_1\cap L_2=\mathbb Q(\zeta_3).
$$