Schanuel's Conjecture and Free Exponential Rings

In this note I show that Schanuel’s Conjecture implies that the exponential subring of R generated by 1 is free on no generators. The interpretation of this result is that Schanuel’s Conjecture, which does not make explicit mention of iterated exponentials, in fact implies that there are no hidden iterated exponential identities for exponential constants. I claim no real originality for the methods used, but I believe that the result in the above suggestive form has not previously been noted. I got the idea from a preprint of Shackell [8] (clearly inspired by Lang’s [6]). I was then unaware of the earlier [l], which, like Shackell’s paper, is concerned with the decision problem for exponential constants. I will touch on this issue at the end of the paper.