References on $\pi^{n}$ is transcendental

Question

I am looking on references showing that $\pi^{n}$ is transcendental for $n\in\mathbb{N}$. Which one do you recommend?

Thanks a lot.

Answer 1

If $\pi^n$ were algebraic, for some $n\ne 0$, then there would an $m\in\mathbb N$ and $a_0,\ldots,a_m\in\mathbb Z$, with $a_ma_0\ne 0$, such that $$ a_m (\pi^n)^m+a_{m-1} (\pi^n)^{m-1}+\cdots+a_0=0, $$ but that would mean that $\pi$ is also algebraic!