Question

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),

1. 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?

2. 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)

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

Answer 1

For (2), the answer is no, not in general. Here is a simple example: take $K_1=\mathbb{Q}(i)$ and $K_2=\mathbb{Q}(\sqrt{-5})$. Then both $K_1/\mathbb{Q}$ and $K_2/\mathbb{Q}$ are (totally) ramified at $p=2$ and $K_1\cap K_2=\mathbb{Q}$, but $F=K_1K_2$ is not totally ramified at $2$. In other words, the extension $\mathbb{Q}(\sqrt{-5},i)/\mathbb{Q}(\sqrt{-5})$ is unramified at $2$ (in fact, $F$ is the Hilbert class field of $K_2$, so it is unramified everywhere).

On the other hand, if you take $K_1=\mathbb{Q}(i)$ and $K_2=\mathbb{Q}(\sqrt{2})$, then $F=K_1K_2$ is totally ramified at $2$ over $\mathbb{Q}$. (Here $F=\mathbb{Q}(\zeta_8)$.)

In both cases, $I_1=D_1=G_1$ and $I_2=D_2=G_2$ (in your notation) but in the first case the inertia in the compositum has order $2$ and in the second case it has order $4$. This shows that one needs to know more than the decomposition and inertia subgroups at a prime in each $K_i$ to understand the ramification index in the compositum.