  Axiomatizing mathematical structures is a goal of Mathematical Logic.
Axiomatizability of the theories of some structures have turned out to be quite
difficult and challenging, and some remain open. However axiomatization of some
mathematical structures are now classical theorems in Logic, Algebra and
Geometry. In this paper we will study the axiomatizability of the theories of
multiplication in the domains of natural, integer, rational, real, and complex
numbers. We will review some classical theorems, and will give some new proofs
for old results. We will see that some structures are missing in the
literature, thus leaving it open whether the theories of that structures are
axiomatizable (decidable) or not. We will answer one of those open questions in
this paper.
