Schanuel's Conjecture Implies Algebraic Independence of Pi and Euler's Number over the Rationals

Theorem
Let Schanuel's Conjecture be true.

Then $\pi$ and $e$ are algebraically independent over the rational numbers $\Q$,

where $\pi$ denotes pi and $e$ denotes Euler's number.

Proof
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 over $\Q$.

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