Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, Galois Theory and the basic theory of transcendental extensions. This is the problem:
Let $W$ be the set of prime numbers and let $\mathfrak{U}$ be a non principal ultrafilter over $W$. For every prime $p$, let $\overline{\mathbb{F}_p}$ denote an algebraic closure of $\mathbb{F}_p$. In the following product $$ \prod_{p\in W} \overline{\mathbb{F}_p}, $$ define the equivalence relation $(a_p)\sim (b_p)$ if and only if $\{p\in W: a_p=b_p\}$ belongs to $\mathfrak{U}$.
Let $$ A=\prod_{p \in W} \overline{\mathbb{F}_p} ~/~ \sim $$ denote the set of equivalence classes, such that $[a_p]$ is the class of $(a_p)$.
- Show that $A$ is not countable.
- Show that $A$ is isomorphic to $\mathbb{C}$.
Proving that the addition and product ($[a_p]+[b_p]=[a_p+b_p]$, $[a_p][b_p]=[a_pb_p]$) are well defined on $A$ and that $\text{char}(A)=0$ is tedious but quiet straightforward. On the other hand proving (1) and (2) has been a rather difficult task. For (1), I honestly have no idea how to proceed; and for (2) I've tried using the fact that $\mathbb{C}$ has an infinite number of automorphism, but I haven't had any luck. Also I think that in order to prove (2) I actually need to show that $|A|=2^{\aleph_0}$.
Any suggestions on how to proceed will be really appreciated. Thanks in advance for any help you provide.