Abstract
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, e x ) modulo Schanuel's Conjecture. We prove that there are no further relations between π and i assuming Schanuel's Conjecture except the known ones, e πi = −1 and i 2 = −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.
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