Intersection of Decomposition groups

Question

Let $K/{\Bbb Q}$ be a finite Galois extension. Choose a prime $p$ and consider the set of primes ${\frak P}_1, \ldots, {\frak P}_n$ of ${\cal O}_K$ lying over $p$. Let $D_{{\frak P}_i} \colon= \{\sigma \in {\mathrm{Gal}}(K/{\Bbb Q})\,|\,\sigma({\frak P}_i) = {\frak P}_i\}$ be the decompositon group of ${\frak P}_i$.

Q. What is known about the intersection $I_p \colon= \underset{i = 1,\,\ldots\,,n}{\bigcap}\!\!\! D_{{\frak P}_i}$ ?

I would like to know what $I_p$ contains as number theoretic information. When $I_p = e$?

Answer 1

All I can say is the following rather banal observation. It won't fit into a comment, so an answer it is.

If $\tau(\mathfrak{P}_i)=\mathfrak{P}_j$ then $D_{\mathfrak{P}_i}=\tau^{-1}D_{\mathfrak{P}_j}\tau$. As the Galois group $G=Gal(K/\Bbb{Q})$ acts transitively on the set of prime ideals above $p$ we arrive at the conclusion that $I_p$ is the so called core of $D_{\mathfrak{P}_i}$. In other words, it is the largest normal subgroup of $G$ contained in $D_{\mathfrak{P}_i}$.

$I_p$ is trivial if and only if there are no non-trivial normal subgroups of $G$ inside $D_{\mathfrak{P}_i}$.

We have the following extremal cases:

• If $G$ is abelian, then $I_p=D_{\mathfrak{P}_i}$ for all $i$ as the decomposition group is independent of the choice of prime ideal above $p$.
• If $G$ is simple and non-abelian, then $I_p$ will always be trivial. It is known that all the decomposition groups are solvable, hence their intersection is a proper subgroup of $G$ in this case. Given that $I_p\unlhd G$, simplicity implies the claim.