Let $\Bbb Q \neq K' \subset K$ be number fields such that $K/\Bbb Q$ and $K'/\Bbb Q$ are Galois. Let $P$ be a prime ideal of $\mathcal O_K$.
Is it true that $$ P^r \cap K' = (P \cap K')^r $$ for any $r \geq 1$?
I know that $\supset$ hold very generally. I tried to factor $ P^r \cap K'$ into a product of prime ideals in $K'$, without success.
Thank you!