Field of algebraic numbers over Q with p-adic value - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T18:48:40Z http://mathoverflow.net/feeds/question/63845 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63845/field-of-algebraic-numbers-over-q-with-p-adic-value Field of algebraic numbers over Q with p-adic value Beatty 2011-05-03T20:30:42Z 2011-05-03T20:30:42Z <p>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 $\overline{\mathbb{Q}}$.</p>