TITLE: Splitting of a prime in compositum
QUESTION [8 upvotes]: Let $K$ be a number field. Let $L_1$ and $L_2$ be two extensions of $K$. Let $P\subseteq O_K$ be a prime ideal. We know that if $P$ is unramified in $L_1$ and $L_2$ it remains unramified in the compositum $L_1L_2$.
Do we have a similar theorem when $P$ splits completely in both $L_1$ and $L_2$, namely: 
If $P$ splits completeely in both $L_1$ and $L_2$, can we say that $P$ splits completely in the compositum $L_1L_2$?

REPLY [5 votes]: @Gal Porat. It is perhaps more "visible" to use the completion of the number field $K$ at $P$. The prime $P$ splits completely in $L_i$ if and only if the local field $K_P$ coincides with $(L_i)_Q$ for all primes $Q$ of $L_i$ above $P$. Hence if $P$ splits completely in both $L_1$ and $L_2$, it will split completely in their compositum.