Assume that $K$ be a number field and $L/K$, $L'/K$ are two separable extensions. Now let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$. Then if $\mathfrak{p}$ is totally split ind $L$ and $L'$, why is $\mathfrak{p}$ also a totally split in $LL'$?
My approach is to prove that every prime ideal $\mathfrak{P}$ of $L$ or $L'$ over $\mathfrak{p}$ is totally split in $LL'$, but I couldn't prove this either.
