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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?

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up vote 12 down vote accepted

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”.

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In particular, from the second paragraph, the weaker question asking whether every field of characteristic zero has an ultrapower which is isomorphic to an ultraproduct of fields of positive characteristic is also negative. – François G. Dorais Jan 30 '14 at 22:18

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