If $\lambda '$ is a root of $\chi$ then $\mathbb{Q}(\xi,\lambda ')$ has degree at most 2 over $\mathbb{Q}(\xi)$ so $\lambda '$ would have to be either a power of $\xi$ or a primitive $4n$-th root of unity. But the minimal polynomial of the latter is given by $\lambda^2-\xi$ hence this is not the case. So what remains to show is that no power of $\xi$ is a root of this polynomial.