How many countable non-isomorphic models does $\text{ACF}_p$ have?

Question

I know $\text{ACF}_p$ is $\kappa$ categorical for all uncountable $\kappa$, but I can't find the number of countable non-isomorphic models of it. I think since the number of prime numbers is countable it should be at least countable.

Answer 1

Two algebraically closed fields are isomorphic iff they have the same characteristic and transcendence degree. For countable fields, the transcendence degree can be any countable cardinality. So there are countably many countable models of $ACF_p$ up to isomorphism, one for each trascendence degree from $0$ to $\aleph_0$.

Note that $ACF_p$ is the theory of algebraically closed fields of a fixed characteristic $p$, so we do not get different models for different $p$ (if we change $p$, we are changing the theory).