# Schanuel's Conjecture Implies Transcendence of Pi plus Euler's Number

## Theorem

Let Schanuel's Conjecture be true.

Then $\pi + e$ is transcendental.

## Proof

Assume the truth of Schanuel's Conjecture.

That is, no non-trivial polynomials $\map f {x, y}$ with rational coefficients satisfy:

$\map f {\pi, e} = 0$

Aiming for a contradiction, suppose $\pi + e$ is algebraic.

Then there would be a non-trivial polynomial $\map g z$ with rational coefficients satisfying:

$\map g {\pi + e} = 0$

However, $\map f {x, y} := \map g {x + y}$ would be a non-trivial polynomial with rational coefficients satisfying:

$\map f {\pi, e} = 0$

which contradicts the earlier statement that no such polynomials exist.

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

$\blacksquare$