Define $\overline{\mathbb{Q}} \subset \mathbb{C}$ to be the subset consisting of all complex numbers which are algebraic over $\mathbb{Q}$. We know that $\overline{\mathbb{Q}}$ is a countable field and that is algebraically closed.
1. Show that there exists a sequence of finite extensions $E_{0}=Q \subset E_{1} \subset \ldots \subset E_{n} \subset \ldots \overline{\mathbb{Q}}$, i.e. each $E_{i}/E_{i-1}$ is a finite exntesion and $\overline{\mathbb{Q}} = \cup_{n} E_{n}$.
2. (Using the above) show that for any prime $p$, the $p$-adic absolute value extends to an absolute value on $\overline{\mathbb{Q}}$.
