This question is related to Riemann's Existence Theorem as cited in Neukirch, Schmidt et. al.: Cohomology of Number fields (10.5.1).
Let $k$ be a number field, $T \supseteq S \subseteq S_p \cup S_\infty$ be a set of primes of $k$ and $p$ a prime number. Then the canonical homomorphism
$$\Phi_{T,S}: \ast _{\mathfrak{p} \in T \setminus S (k_S(p))} T_\mathfrak{p}(k(p)\mid k) \to G(k_T(p)\mid k_S(p))$$
is an isomorphism.
How can I see, that this homomorphism is canonically surjective?
Notations:
- $k_T(p)$ means the maximal pro-$p$-extension of $k$ that is unramified outside the set $T$
- $T_\mathfrak{p}(k(p)\mid k)$ denotes the inertia group of $\mathfrak{p} \in k(p)$
- $G(k_T(p) \mid k_S(p))$ is the Galois group of the extension $k_T(p) \mid k_S(p)$
- $T \setminus S(k_S(p))$ are the primes of $T \setminus S$ on the level of $k_S(p)$
Thank you!