I have two extensions of the algebraic numbers, and I'd like to know whether they are equivalent.
Definition 1. $x \in \mathbb E_1$ iff $x$ is a root of $e^{P(x)} + Q(x)$ with $P, Q$ some polynomials with integer coefficients.
Definition 2. $x \in \mathbb E_2$ iff $x$ is a root of $e^{P(x)} + Q(x)$ with $P, Q$ some polynomials with integer complex coefficients. (Here, integer complex coefficients are numbers in the form $a+bi$, with $a,b \in \mathbb Z$.)
It is easy to see they are both countable, so we can't get a contradiction form there.
I don't think they are equivalent, because with definition 2 we can make $\pi$ since it is a root of $e^{ix} +1$.
Question: Are $\mathbb E_1$ and $\mathbb E_2$ the same set?