[2] To see this first note that for any element x in a field with characteristic p > 0, Hence, taking into account the Frobenius endomorphism, And so E(x) = 1 for every x.
[3] The underlying set F may not be required to be a field but instead allowed to simply be a ring, R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group in R to the multiplicative group of units in R. The resulting object is called an exponential ring.
It was proved in the 1990s that Rexp is model complete, a result known as Wilkie's theorem.
This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that Rexp is also o-minimal.
It is known that if the real version of Schanuel's conjecture is true then Rexp is decidable.