Fields of characteristic zero via ultraproducts

Question

Is every noncountable field of characteristic zero the ultraproduct (using a non principal ultrafilter over the set of prime numbers) of fields of positive characteristic?

Answer 1

No. Ultraproducts over nonprincipal ultrafilters on a countable index set are always $\aleph_1$-saturated. This rules out many fields just on cardinality basis (the cardinality of the ultraproduct must satisfy $\kappa=\kappa^\omega$), but even if the field has cardinality $2^\omega$, it does not have to be $\aleph_1$-saturated.

Even more importantly, it is not true that every field of characteristic $0$ is elementarily equivalent to an ultraproduct of fields of nonzero characteristic. In other words, there exist first-order sentences that are satisfiable in a field of characteristic $0$, but not in any field of positive characteristic. One such sentence is “every sum of two squares is a square, and $-1$ is not a square”.