In recent years Schanuel’s Conjecture (SC) has played a fundamental role
in the Theory of Transcendental Numbers and in decidability issues.
Macintyre and Wilkie proved the decidability of the real exponential field,
modulo (SC), solving in this way a problem left open by A. Tarski.
Moreover, Macintyre proved that the exponential subring of R generated
by 1 is free on no generators. In this line of research we obtained that in
the exponential ring $(\mathbb{C}, ex)$, there are no further relations except $i^2 = −1$
and $e^{i\pi} = −1$ modulo SC. Assuming Schanuel’s Conjecture we proved that
the E-subring of $\mathbb{R}$ generated by $\pi$ is isomorphic to the free E-ring on $\pi$.
These results have consequences in decidability issues both on $(\mathbb{C}, ex)$ and
$(\mathbb{R}, ex)$. Moreover, we generalize the previous results obtaining, without
assuming Schanuel’s conjecture, that the E-subring generated by a real
number not definable in the real exponential field is freely generated. We
also obtain a similar result for the complex exponential field.