If $\pi$ is not algebraic number then : is $\pi ^{n}$ algebraic number for $n >1$?

Question

if $\pi$ is not algebraic number then : is $$\pi ^{n}$$ algebraic number for $n >1$ ?

Thank you for any kind of help .

Answer 1

Assume that $\pi^n$ is algebraic, with minimal polynomial $p(x)$.

Then $\pi$ is a root of $p(x^n)$, hence $\pi$ is algebraic, contradiction.