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.
- 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 $\overline{\mathbb{Q}} = \cup_{n} E_{n}$.
- (Using the above) show that for any prime $p$, the $p$-adic absolute value extends to an absolute value on $\overline{\mathbb{Q}}$.
So, I proved 1 (just define $E_{i} = \mathbb{Q} (a_{1}, a_{2},\ldots, a_{i}$, where $\overline{\mathbb{Q}} = \left\{a_{1}, a_{2},\ldots\right\}$ ), but I don't know how to formalize $2$. Of course, using the fact that every nonarchimedean absolute value on a field, extends in at least one way to every finite extension, we get an extension of the p-adic valuation on each $E_{n}$, but I don't see how to end $2$. Perhaps, a continuity argument?
Thanks.