Find natural numbers $n, m \in \mathbb N, n$ odd, in which $\sqrt[n]m\notin\mathbb N$ but $x^n-m$ is not minimal polynomial in $\mathbb Q[x]$

Question

Find natural numbers $n,m \in \mathbb N$, $n$ odd, where $\sqrt[n]m\notin \mathbb N$, but $x^n-m$ is NOT minimal polynomial in $\mathbb Q[x]$ of unique real root $r=\sqrt[n]m\in \mathbb R$.

My attempt:

I have no idea where to start. I realized that roots of $x^n-m$ are exactly:

$$r, re^{i\frac{2\pi}{n}}, ..., re^{i\frac{2(n-1)\pi}{n}},$$

where all roots except the first one are complex. Is there a way to make up another polynomial $f(x)\in \mathbb Q[x]$ with some of the complex roots above? Then we can reduce $x^n-m$ with rational coefficients.

Answer 1

The minimal polynomial of $\sqrt[9]{27}$ over the rationals is $x^3-3$.