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

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

• It's worth pointing out that the normal closure of $\mathbb{R}\cap\bar{\mathbb{Q}}$ over $\mathbb{Q}$ is $\bar{\mathbb{Q}}$ (because it contains the three roots of $t^3-2$, so it contains their quotients, i.e., the primitive cube roots of unity, which gives us what we need to generate $\bar{\mathbb{Q}}$). – Gro-Tsen Feb 14 '17 at 13:40
• 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 ${\rm{Gal}}(K_s/K)$ 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. – nfdc23 Feb 14 '17 at 15:50
• @nfdc23 You should post this as an answer. – Gro-Tsen Feb 14 '17 at 18:17

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.