$ \pi + e $ or $ \pi - e $ is transcendental

Question

I know how to proof that either $ \pi + e $ or $ \pi * e $ is transcendental, but I don't know how to tackle this problem.

We must assume that we don't know if either of the expressions are transcendental. But we have to proof that one of them is.

Answer 1

Similarly to the other argument, if that were not the case, then $x_1=\pi$ and $x_2=e$ would be solutions to the system $$\begin{cases}x_1+x_2=\mathrm{algebraic}_1\\ x_1-x_2=\mathrm{algebraic}_2\end{cases}$$

and thus both $\pi$ and $e$ would be algebraic.