recently I'm taking mathematical logic course, and the class covers some basic model theoretical ideas. Since I have not taken any abstract algebra course, It is so hard to understand what is going on.
proving ACFp is not $w$-cateogircal, professor defined $F_n=Q(\alpha_i |i<n)$ where $\alpha_i$ is transcendental and algebraically independent. he just wrote "if $m\neq n$, then $F_n\ncong F_m$, but $F_n, F_m \models T_{ACF_0}$" where $T_{ACF_0}$ is a set of axioms of algebraically closed field of characteristic 0. I don't get what is $Q(\alpha_i |i<n)$, so do $F_n\ncong F_m$, and $F_n, F_m \models T_{ACF_0}$.