Some explicit axiomatizations for the theories of <C,x>, <R,x> and <Q+,x>.
What is it about?
1- The multiplicative theory of the complex numbers, i.e., the first order theory of the structure (C,x), is decidable by Tarski's Theorem. Here we prove this result directly, and present an explicit axiomatization for it. 2- The multiplicative theory of the real numbers, i.e., the first order theory of the structure (R,x), is decidable by Tarski's Theorem. Here we prove this result directly, and present an explicit axiomatization for it. 3- The multiplicative theory of the positive rational numbers, i.e., the first order theory of the structure (Q+,x), is decidable by Mostowski's Theorem. Here we prove this result directly, and present an explicit axiomatization for it.
Why is it important?
Axiomatizing mathematical theories is an important goal of mathematical logic. Here we have achieved this goal for the multiplicative theories of the complex, real and (positive) rational numbers.