I'm currently self studying Ireland and Rosen's A Classical Introduction to Modern Number Theory and got stuck on the proof of Proposition 13.2.9. In this proof, $p$ is a prime not dividing $m$, $D, D_m$ denote the ring of algebraic integers in $\mathbb{Q}(\zeta_p,\zeta_m)=\mathbb{Q}(\zeta_{pm})$ and $\mathbb{Q}(\zeta_m)$ respectively (where $\zeta_m$ is the primitive mth root of unity). The authors have shown that $pD=(P_1P_2...P_{g'})^{e'(p-1)}$ where $P_i$ are distinct prime ideals of degrees $f'$. On the other hand it is shown that $pD_m=\widetilde{P_1}\widetilde{P_2}...\widetilde{P_g}$ where $\widetilde{P_i}$ are distinct prime ideals of degrees $f$. The authors then claimed "by considering the prime decomposition of $\widetilde{P_i}D$ and comparing the equations we see $f' \geq f$ and $g' \geq g$". However I cannot see why the first inequality should hold.
Would anyone please help me out with this part of the proof? Thank you very much in advance.
After some time I think I came up with a valid reason, I'll post it here for others' reference: Consider the mapping $d+\widetilde{P_i} \mapsto d+P_j$, from $D_m/\widetilde{P_i}$ to $D/P_j$ where $P_j$ is a prime ideal factor of $\widetilde{P_i}D$. One can show this mapping is well-defined and injective. Thus $p^f=|D_m/\widetilde{P_i}| \leq |D/P_j|=p^{f'}$.