I'm studying pseudo-finite fields in particular on Chatzidakis notes. When dealing with the concept of pseudo-algebraically closed (PAC) fields, it is stated that they cannot be finite but without any proof. I've tried to figure out a proof by myself trying to find an absolutely irreducible polynomial over a generic finite field $F_q$ and show it doesn't admit an $F_q$-rational point but I suspect this cannot work. Does anyone have a better idea?
2026-03-27 02:39:58.1774579198
Proof that finite fields cannot be pseudo-algebraically closed
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For $2\nmid q$ I have the following example: $y^2 = x^q - x + d$, where $d\in \Bbb F_q$ is not a square.
It is easy to see that this equation defines an absolutely irreducible variety and has no $\Bbb F_q$ point.
I believe you can also modify it to an example in char $2$.