Suppose $\alpha$ is a real algebraic number with the property that its irreducible polynomial over $\mathbb{Q}$ is not a binomial, i.e., it is not of the form $x^n-q$ for some $n\geq 1$ and $q\in\mathbb{Q}$.
True or false: $\alpha^k\not\in\mathbb{Q}$ for all $k\geq 1$.
Example: $\sqrt{2}+\sqrt{3}$ has irreducible polynomial $x^4-10x^2+1$, and no integer power of $\sqrt{2}+\sqrt{3}$ is rational.
Non-example when "real" assumption is dropped: $1+i$ has irreducible polynomial $x^2-2x+2$, but $(1+i)^4=-4\in\mathbb{Q}$.