What is the normal closure of $\mathbb{Q}_p \cap \bar{\mathbb{Q}}$ over $\mathbb{Q}$? Is it $\bar{\mathbb{Q}}$?

(Copied from the @nfdc23's comment, because it's an answer.)

For any global field $K$, place $v$ of $K$, and embedding of $K_s$ into a separable closure of $K_v$, the normal closure of $K_v \cap K_s$ over $K$ is $K_s$. Indeed, such an intersection corresponds to the decomposition group of $\operatorname{Gal}(K_s/K)$ at a place of $K_s$, and the intersection of decomposition groups for even just two distinct places of $K_s$ (such as two over a place of $K$) is trivial. See 12.1.3 (and 12.1.9 and 12.1.11) in the book *Cohomology of Number Fields* by Neukirch, Schmidt, and Wingberg.

Cohomology of Number Fieldsby Neukirch, Schmidt, and Wingberg. $\endgroup$ – nfdc23 Feb 14 '17 at 15:50