Let $\mathbb{N}^*$ be a countable non-standard model of Peano arithmetic (PA) and let $\mathbb{Z}^*$ be the integers extended from $\mathbb{N}^*$. A non-standard finite field would be a ring $\mathbb{Z}^* /n^* \mathbb{Z}^*$ where $n^*$ is a non-standard prime number larger than any standard natural number.
How is a nonstandard finite field different from a pseudo-finite field? Can there be a mapping from one to the other?
Answer
The answer is negative. Not every pseudo-finite field, i.e a model of the theory of finite fields, is "nonstandard integers modulo a nonstandard prime". Every field of this form has characteristic zero. By contrast, there are pseudo-finite fields of positive characteristic:
http://www.logique.jussieu.fr/~zoe/papiers/Helsinki.pdf
(see page 17, example 5.1)
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