What are some examples of non isomorphic countable algebraically closed fields of characteristic zero? Or they don´t exist

Question

I was reading some model theory, and saw stated that $ACF_p$ (the first order theory of algebraically closed fields of characteristic $p$) is $\kappa$-categorical for all $\kappa > \aleph_0$. Is this result is false for $\kappa=\omega$?
In case that $ACF_p$ isn't $\omega$-categorical, this would mean that there exist at least two countable algebraically closed fields of characteristic $0$ that are not isomorphic. What would be some example of such fields? do we know all of them? (up to isomorphism, of course).
In case that $ACF_p$ is indeed $\omega$-categorical, how would a proof of that look like? The one provided for the case $\kappa>\aleph_0$ relies heavily on uncountability.
Thanks in advance to anyone who answers!

Answer 1

Algebraically closed fields are determined up to isomorphism by their characteristic $c$ and their transcendence degree (over their prime subfield, i.e. over $\Bbb Q$ if $c=0$ and over $\Bbb Z_c$ if $c\ne0$). Hence the countable algebraically closed fields of characteristic $0$ are, up to isomorphism: $$\overline{\Bbb Q(X_k,k\in F)}\quad F\text{ at most countable,}$$ where $\Bbb Q(X_k,k\in F)$ denotes the field of rational functions with rational coefficients and set of indeterminates $\{X_k\mid k\in F\}.$