TITLE: Irrationality of powers of $\pi$
QUESTION [14 upvotes]: Everyone knows that $\pi$ is an irrational number, and one can refer to this page for the proof that $\pi^{2}$ is also irrational.
What about the highers powers of $\pi$, meaning is $\pi^{n}$ irrational for all $n \in \mathbb{N}$ or does there exists a $m \in \mathbb{N}$ when $\pi^{m}$ is rational.

REPLY [31 votes]: What Robin hinted at:
If $\pi^{n}$ was rational, then $\pi$ would not be transcendental, as it would be the root of $ax^{n}-b = 0$ for some integers $a,b$.