I know, that field ${\mathbb{Q}_p}$ (field of p-adic numbers) has the same cardinality as $\mathbb{C}$. Taking algebraic closure doesn't change the cardinality of infinite field, so cardinality $\overline{{\mathbb{Q}}_p}$ is also equal to ${|\mathbb{C}|}$. Why taking completion (passing to $\mathbb{C}_p : = \widehat{\overline{\mathbb{Q}}_p}$) doesn't change the cardinalty? I require this step in order to prove the isomorphism of the previous field to the field of complex numbers.
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6$\begingroup$ The completion of a metric space of cardinal $2^{\aleph_0}$ is $2^{\aleph_0}$. This is just because $(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}$ and elements of the completions are limits of converging sequences from the smaller space. $\endgroup$– YCorCommented Apr 16, 2017 at 11:20
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Not only does $\mathbb C_p$ have the same cardinality as $\mathbb C$, but the larger field $\Omega_p$, the spherical completion of $\overline{\mathbb Q}_p$, also has this cardinality. Further, one can explicitly describe $\Omega_p$ as the set of series $$\sum_{r\in\mathbb Q} c_rp^r$$ with coefficients given by Teichmüller representatives of $\overline{\mathbb{F}}_p$, such that the set of exponents with nonzero coefficients forms a well-ordered subset of the rationals. (I'm not sure to whom this description of $\Omega_p$ is due, but it is in Bjorn Poonen's undergraduate thesis.)
answered Apr 16, 2017 at 11:17
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$\begingroup$ Is not $\mathbb{C}_p=completion \ \ of \ \ \bar{\mathbb{Q}}_p=\Omega_p$ ? $\endgroup$– MASCommented Feb 15, 2019 at 9:39
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2$\begingroup$ @M.A.SARKAR The completion of $\overline{\mathbb Q}_p$ with respect to its metric is $\mathbb C_p$, but $\mathbb C_p$ is not sperically complete, which is a different concept.. (Spherical completness means that any nested decreasing sequence of closed balls has non-empty intersection. $\mathbb C_p$ does not have this property, hence the Type IV points in Berkovich space.) The spherical completion is a smallest field that is spherically complete. $\endgroup$ Commented Feb 15, 2019 at 11:18
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$\begingroup$ Much of the general theory of spherically complete fields was developed in the 1930s and 1940s by Schmidt, Krull, Ostrowski, and especially Kaplansky in his thesis doi.org/10.1215/S0012-7094-42-00922-0 . The explicit description of the field of transfinite series is due to Hahn in 1908 in the equicharacteristic case and to Lampert doi.org/10.1016/0022-314X(86)90073-9 in 1986 in the $p$-adic case above. $\endgroup$ Commented Jul 30, 2021 at 22:39
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$\begingroup$ What my undergraduate thesis (Poonen, Maximally complete fields, Enseign. Math. (2) 39 (1993), no. 1-2, 87-106) did was to generalize the $p$-adic construction to arbitrary value groups and residue fields, to prove that $\Omega_p$ and these generalizations are maximally complete (hence spherically complete, hence the same as Kaplansky's fields), and to prove various other statements - for instance, that the transcendence degree of $\Omega_p$ over $\mathbb{C}_p$ is $2^{\aleph_0}$. $\endgroup$ Commented Jul 30, 2021 at 22:49
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$\begingroup$ (A valued field $(K,v)$ is maximally complete if the only valued field extension of $(K,v)$ having the same value group and same residue field is $(K,v)$ itself. Kaplansky proved that this is equivalent to being spherically complete.) $\endgroup$ Commented Jul 30, 2021 at 22:56