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.



Cardinality of ${\mathbb{C}_p}$
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.)


