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



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


