Is the theory of real closed fields finitely axiomatizable?

Question

I don't know if this has been asked before, but is the theory of real closed fields $(F;+,-,*,0,1,<)$ finitely axiomatizable? The usual axiomatization is to take the axioms for ordered fields and add an axiom stating that every positive number has a square root and an axiom schema stating that every odd-degree polynomial has a root. I am wondering if this axiomatization can be replaced by a finite one, and if not, the proof that it can not.