Proof $\pi*e$ is transcendental.
either $\pi + e$ or $\pi*e$ is transcendental to see take $(x-\pi)(x-e)=x^2-(\pi+e)x+\pi*e$.
Case 1 assume $\pi$ and $e$ are algebraically independent. It follows immediately that $\pi*e$ is transcendental.
Case 2 assume $\pi$ and $e$ are algebraically dependent. It follows for some polynomial with integer coefficients that $p(\pi,e)=0$, it follows that ... Then "$(\pi +e)$" is algebraic. Then $\pi*e$ is transcendental.
by both cases $\pi*e$ is transcendental.
Or do you think this approach will be ultimately fruitless?
Also I'm not the best at latex so if someone wants to edit this to make it proper I would be grateful.