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The ultraproduct of all finite prime fields $ \Bbb F_p $ (over a nonprincipal ultrafilter U) is a field of characteristic 0. How do I show that it has exactly one extension of degree n for each natural number n?

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Hint: First show that this is true for all of the fields $\mathbb{F}_p$.

Then (Harder) try to express this in a first-order way.

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  • $\begingroup$ To help Dominik: For the second part, things would be easier to do if one could refer to elements of an algebraic extension directly. But you can! Simply consider the ultraproduct of the structures $(\bar{\mathbb F_p},\mathbb F_p,\dots)$ to begin with. $\endgroup$ – Andrés E. Caicedo Jun 14 '13 at 1:51

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