I know that there are uncountably infinite transcendentals over $\mathbb Q$ in $\mathbb Q_p$. What i want to ask is if there is a way to determine whether a transcendental over $\mathbb Q$ lays in the field $\mathbb Q_p$. Thank you!
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4$\begingroup$ One possible misconception here is that $\mathbf Q_p$ is naturally a subfield of $\mathbf R$ (or $\mathbf C$) and the question is asking something like "how to tell if a transcendental real is in $\mathbf Q_p$". I could be completely wrong, of course. $\endgroup$– user5117Commented Jun 25, 2014 at 13:12
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2$\begingroup$ Does it make sense to ask whether $\pi$ is in $\mathbb Q_p$? $\endgroup$– Gerald EdgarCommented Jun 25, 2014 at 13:14
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5$\begingroup$ First, this is not the right level question for this site... but it is an interesting and deceptive sort of issue. I read somewhere that, for example, Hensel marred his own reputation by giving an irremediably flawed proof that $\pi$ is irrational by accidentally assuming that a sequence of rational numbers that is Cauchy in $\mathbb R$ and in $\mathbb Q_p$ has a limit that is "the same" in both, in some alleged sense... which is not the case. $\endgroup$– paul garrettCommented Jun 25, 2014 at 13:32
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5$\begingroup$ It does not make sense to ask if $\pi$ is in $\mathbb{Q}_p$. Even in situations where you are tempted to say a certain real number is in $\mathbb{Q}_p$ (e.g. $\sum p^n/n!$ converges $p$-adically and defines a transcendental number) it is not correct as there is no canonical way to embed $\mathbb{Q}_p$ in $\mathbb{C}$. In particular, my parenthetical number is not $e^p$ in any meaningful way. This question is based on a misunderstanding of the definitions and should be closed. $\endgroup$– Felipe VolochCommented Jun 25, 2014 at 14:23
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2$\begingroup$ It seems to me that one way to turn the question into a meaningful one (although may be uninteresting to the OP) is to ask whether there exists an explicit way to tell whether an element of $\mathbb{C}_p$ (or of $\overline{\mathbb{Q}_p}$ or of any field $K\supseteq\mathbb{Q}_p$) which is trascendental over $\mathbb{Q}$ is already in $\mathbb{Q}_p$: like testing whether a trascendental complex number is real but in the $p$-adic world. If $K/\mathbb{Q}_p$ is not Galois, it seems interesting. $\endgroup$– Filippo Alberto EdoardoCommented Jun 25, 2014 at 22:19
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