Let $G\_{\mathbb{Q}\_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}\_{p}$. Also, their structure as abstract groups is completely known.


It is well known that this group embeds into the absolute Galois group of the rationals, $G\_{\mathbb{Q}}$.


My question is: are there any known relations between the two absolute Galois groups of two $p$-adic fields, say $\mathbb{Q}\_{p\_1}$ and $\mathbb{Q}\_{p\_2}$, where $p\_1$ and $p\_2$ are different primes, regarded as subgroups of the absolute Galois group of the rationals?



Relation between $G_{\mathbb{Q}_p}$ for different primes
As noted in the remarks above, there are different (although conjugate) embeddings $G\_{\mathbb{Q}\_p}\rightarrow G\_\mathbb{Q}$. Let us fix one of them and denote its image by $G\_p$.


If we fix prime numbers $p\_1,\dots,p\_n$ (not necessarily distinct), then
the set of $\sigma=(\sigma\_1,\dots,\sigma\_n)\in G\_\mathbb{Q}^n$ for which the closed subgroup
$\left< G\_{p\_1}^{\sigma\_1},\dots,G\_{p\_n}^{\sigma\_n}\right>$ is the profinite free product of $G\_{p\_1},\dots,G\_{p\_n}$ has Haar measure $1$.
This is (a special case of) Theorem 4.1 in W.-D. Geyer, Galois groups of intersection of local fields, Israel Journal of Mathematics 30, 1978.


Geyer also notes in section 4.4 that one cannot expect this to hold for *every* $\sigma$, but his example is for archimedean places. A purely group theoretic result of Haran occurring as Proposition 4.2.3 in Neukirch-Schmidt-Wingberg's Cohomology of Number Fields however shows that also for non-archimedean places it does not hold for *every* $\sigma$.


