User b rosenfield - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:38:07Z http://mathoverflow.net/feeds/user/4512 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17032/which-p-adic-numbers-are-also-algebraic/17656#17656 Answer by B Rosenfield for Which p-adic numbers are also algebraic? B Rosenfield 2010-03-09T21:36:03Z 2010-03-09T21:36:03Z <p>There's a slightly subtle point near here of which some people are not aware: that it is dangerous (perhaps even nonsensical) to compare algebraic numbers under various different completions. So, to talk about $Q_p\cap \bar Q$, you should be talking about a completion of $Q$ containing $Q_p$, not, e.g., a completion of $Q$ lying inside $C$. I don't think this is what is happening here, but some people may find this interesting.</p> <p>Now, there are lots of isomorphisms floating around, so usually everything turns out just fine, but sometimes not. Here are two examples.</p> <p>(1) The following fallacious argument that $e$ is transcendental is from a talk by Gouvêa, "Hensel's p-adic Numbers: early history" (originally due to Hensel himself).</p> <p>The series expansion of $e^p$ converges in $Q_p$, thus $e$ is a solution to the equation $X^p=1+p\epsilon$, where $\epsilon$ is a $p$-adic unit. So $[Q_p(e):Q_p]=p$ (of course you need to argue that the polynomial is irreducible), and so $[Q(e):Q]\ge p$. Since $p$ was arbitrary, $e$ must be transcendental over $Q$. </p> <p>The fallacy is that even though the series for $e$ (and $e^p$) converges in $R$ and $Q_p$, the numbers they converge to are not the same. </p> <p>(2) The following is from Koblitz's $p$-adic book, page 83 (with an example and some other fallacious arguments).</p> <p>It is <em>not true</em> that if an infinite sum of rational numbers (a) converges $p$-adically to a rational number for some $p$ and (b) converges in the real topology to a rational number, then the rational numbers the two series converge to are the same!</p>