Is the non-principal ultraproduct of finite fields $\prod\_p \mathbb{F}\_p/\sim$ a nonstandard model of the rationals $\mathbb{Q}$?


EDIT: Can we realize $\mathbb{Q}^\*$ as an ultraproduct?



Ultraproduct $\prod_p \mathbb{F}_p/\sim$ and $\mathbb{Q}^*$
It is easy to see that at least one of $-1,2,-2$ is a square in that field: the set of primes where neither $-1$ nor $2$ is a quadratic residue is contained in the set of primes where $-2$ is a quadratic residue.


