Let $p$ be a regular prime i.e. $p\nmid h_p$, ($h_p$ denotes the class number of $\mathbb{Q}(\mu_p)$). Let $F$ be an abelian number field such that $p\nmid h_F$ and $p\nmid [F:\mathbb{Q}]$.
Do we have $p\nmid h_{F(\mu_p)}$?
I hope it holds, but in general the following is wrong:
Let $K,L$ be two number fields such that $p\nmid h_K h_L$, and $p\nmid [KL:K][KL:L]$, then $p\nmid h_{KL}$.
It's easy to find lots of examples from quadratic fields, for instance, take $p=3, K=\mathbb{Q}(i),L=\mathbb{Q}(\sqrt{23})$. $h_{KL}=3$