Suppose {$K_i/\mathbb{Q}$} is a finite set of finite galois extensions of $\mathbb{Q}$ with Galois groups $G_i$.

Suppose we know the ramifications of $K_i$ quite well (e.g., their decomposition groups, inertia groups at some primes),

What can we say about the ramifications of the compositum field of $K_i$ (e.g., the ramification index, inertia degree of some primes)? Any References?

Particularly, when $K_1\cap K_2=\mathbb{Q}$, we know that $K_1K_2$ has Galois group $G_1\times G_2$. Is the corresponding decomposition group (resp. inertia group) of the form $D_1\times D_2$ (resp. $I_1\times I_2$)? (This is wrong in general, see Álvaro Lozano-Robledo's answer for a counterexample)

How about the case if we remove the requirement that $K_i/\mathbb{Q}$ are Galois?

Classical Theory of Algebraic Numbers, p. 263. $\endgroup$ – Watson Jan 5 '17 at 9:15