An explicit axiomatization of the universal theory of fields

Question

Is the $\forall$-theory of fields finitely axiomatizable? If so, what is an explicit axiomatization of it?

Answer 1

Hint: for any theory $T$ its universal consequences $T_{\forall}$ is precisely the theory of substructures of models of $T$. So to figure out what $\forall$-theory of fields is, you need to figure out which rings can be extended to fields.