Some consequences of Schanuel's Conjecture in exponential rings

In last years Schanuel's Conjecture has played a fundamental role in Transcendental Number Theory and in decidability issues. In this article we investigate algebraic relations among the elements of the exponential field (ℂ, ex) modulo Schanuel's Conjecture. We prove that there are no further relations between π and i assuming Schanuel's Conjecture except the known ones, eπi = -1 and i2 = -1. Moreover, modulo Schanuel's Conjecture we prove that the E-subring of ℝ generated by π is isomorphic to the free exponential ring on π.