  The multiplicative theory of a set of numbers (which could be natural,
integer, rational, real or complex numbers) is the first-order theory of the
structure of that set with (solely) the multiplication operation (that set is
taken to be multiplicative, i.e., closed under multiplication). In this paper
we study the multiplicative theories of the complex, real and (positive)
rational numbers. These theories (and also the multiplicative theories of
natural and integer numbers) are known to be decidable (i.e., there exists an
algorithm that decides whether a given sentence is derivable form the theory);
here we present explicit axiomatizations for them and show that they are not
finitely axiomatizable. For each of these sets (of complex, real and [positive]
rational numbers) a language, including the multiplication operation, is
introduced in a way that it allows quantifier elimination (for the theory of
that set).
