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  • $\begingroup$ What does $\widetilde{\mathbb Q}_p$ mean? $\endgroup$ Commented Apr 26, 2021 at 15:18
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    $\begingroup$ $\mathbb Q_p$ is the field of $p$-adic numbers, $\widetilde{\mathbb Q}_p$ is the algebraic closure of $\mathbb Q_p$, and $\mathbb C_p$ is the completion of $\widetilde{\mathbb Q}_p$ with respect to the $p$-adic valuation. $\endgroup$ Commented Apr 26, 2021 at 15:38
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    $\begingroup$ Ah, I'm used to a bar rather than a tilde for algebraic closure, but I understand that when completion and algebraic closure interact only one can get the bar. Thanks! (I guess I should have figured it couldn't be the completion, since that would give a non-algebraically-closed, hence not isomorphic to $\mathbb C$, field.) $\endgroup$ Commented Apr 26, 2021 at 15:39
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    $\begingroup$ Right. Also note that there is not much point in completing $\mathbb Q_p$, as it is already complete (or rather, it itself arises as a completion of $\mathbb Q$). $\endgroup$ Commented Apr 26, 2021 at 17:00
  • $\begingroup$ Ha, right you are. I'm not sure what I meant. $\endgroup$ Commented Apr 26, 2021 at 17:01