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

Theorem

Let Schanuel's Conjecture be true.

Then $\pi$ (pi) and the logarithm of $\pi$ (pi):

$\ln \pi$

are algebraically independent over the rational numbers $\Q$.

Proof

Assume the truth of Schanuel's Conjecture.

Let $z_1 = \ln \pi$, $z_2 = i \pi$.

Note that $z_1$ is wholly real and $z_2$ is wholly imaginary.

Hence, by Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals, they are linearly independent over $\Q$.

By Schanuel's Conjecture, the extension field $\Q \left({z_1, z_2, e^{z_1}, e^{z_2}}\right)$ has transcendence degree at least $2$ over $\Q$.

That is, the extension field $\Q \left({\ln \pi, i \pi, \pi, e^{i \pi}}\right)$ has transcendence degree at least $2$ over $\Q$.

However, by Euler's Identity, $e^{i \pi} = -1$ is algebraic.

Also, $i \pi$ and $\pi$ are not algebraically independent, as they satisfy $x^2 + y^2 = 0$, where $x = i \pi$ and $y = \pi$.

Therefore, if Schanuel's Conjecture holds, then $\ln \pi$ and $\pi$ are algebraically independent.

$\blacksquare$