Omega Constant is Transcendental

Theorem

The omega constant is transcendental.

Proof

From the definition of omega constant, it is the real number $\Omega$ such that:

$\Omega \, e^\Omega = 1$

where $e$ denotes Euler's number.

Aiming for a contradiction, suppose $\Omega$ is not transcendental.

Hence, by definition, $\Omega$ is algebraic.

Then $e^\Omega$ is also algebraic, because:

$e^\Omega = \dfrac 1 \Omega$

However, by the weaker Hermite-Lindemann-Weierstrass theorem, $e^\Omega$ is transcendental.

From this contradiction it follows that $\Omega$ is transcendental.

$\blacksquare$