One can define fields using a finite list of axioms that quantify over the field itself. However, the obvious way to define algebraically closed fields involves either an infinite list of axioms, or a more logically complex framework where one can quantify over natural numbers. It would be interesting if one could avoid this, and give a finite list of axioms that quantify only over the field, but my guess is that that is impossible. Are there known theorems in that direction?



Logical complexity of algebraically closed fields
From Dirk van Dalen's Logic and Structure: the theory of algebraically closed fields is [not finitely axiomatizable](https://books.google.com/books?id=u0wlXPHATDcC&pg=PA108&lpg=PA108&dq=algebraically+closed+finitely+axiomatizable&source=bl&ots=meDLA-Mu8t&sig=U1zZBmNrOjLCpUgqtWc6yTyKAHw&hl=en&sa=X&ved=0CEAQ6AEwBGoVChMIpaiXvJ_MyAIVSxg-Ch1bTQfA#v=onepage&q=algebraically%20closed%20finitely%20axiomatizable&f=false) (see page 109 and preceding).


