An example is given by any cubic Galois extension of $\mathbb{Q}$, e.g. the extension $\mathbb{Q}(\alpha)$ where $\alpha = \cos \frac{2\pi}{9}$ is a root of $x^3 - 3x + 1$.

This works because if the extension were of the form $K(\beta)$ with $\beta^3 \in \mathbb{Q}$, then since it is Galois it would have to contain a nontrivial cube root of unity, which it obviously doesn't.

An example is given by any cubic Galois extension of $\mathbb{Q}$, e.g. the extension $\mathbb{Q}(\alpha)$ where $\alpha = \cos \frac{2\pi}{9}$ is a root of $x^3 - 3x + 1$.

This works because if the extension were of the form $\mathbb{Q}(\beta)$ with $\beta^3 \in \mathbb{Q}$, then since it is Galois it would have to contain a nontrivial cube root of unity, which it obviously doesn't.