Fix an algebraic closure $\overline{\mathbb Q}$ of the field $\mathbb Q$ of rational numbers. For a prime $p$ let $K_p$ the field of all algebraic elements in ${\mathbb Q}_p$.

- Question: Is $K_p$ normal over $\mathbb Q$?

If so, it defines a unique subfield of $\overline{\mathbb Q}$ which we might want to write as ${\mathbb Q}_p\cap \overline{\mathbb Q}$. Then the second question is:

- Is $\bigcap_p{\mathbb Q}_p\cap \overline{\mathbb Q}=\mathbb Q?$