Schanuel's Conjecture Implies Transcendence of Pi by Euler's Number

Theorem
Let Schanuel's Conjecture be true.

Then $\pi \times e$ is transcendental.

Proof
Assume the truth of Schanuel's Conjecture.

Let $z_1 = 1$ and $z_2 = -i \pi$.

By Schanuel's Conjecture, the extension field $\Q \left({1, -i \pi, e, 1}\right)$ has transcendence degree at least $2$ over $\Q$.

However, $1$ is algebraic.

Therefore $-i \pi$ and $e$ must be algebraically independent.

Thus $\pi$ and $e$ are algebraically independent.

Therefore, if Schanuel's Conjecture holds, $\pi \times e$ is transcendental.