Let $K=\mathbb{Q}(\alpha)$, where $\alpha=\sqrt[3]{10}$, and let $\mathcal{O}_{K}$ be the ring of algebraic integers of $K$. Show that the field polynomial of $r+s \alpha$, with $r$ and $s$ rational, is $(x-r)^{3}-10 s^{3}$.

Let $\beta=\frac{1}{3}\left(\alpha^{2}+\alpha+1\right)$. By verifying that $\beta=3 /(\alpha-1)$ and determining the field polynomial, or otherwise, show that $\beta$ is in $\mathcal{O}_{K}$.

By computing the traces of $\theta, \alpha \theta, \alpha^{2} \theta$, show that the elements of $\mathcal{O}_{K}$ have the form

$$\theta=\frac{1}{3}\left(l+\frac{1}{10} m \alpha+\frac{1}{10} n \alpha^{2}\right)$$

where $l, m, n$ are integers. By further computing the norm of $\frac{1}{10} \alpha(m+n \alpha)$, show that $\theta$ can be expressed as $\frac{1}{3}(u+v \alpha)+w \beta$ with $u, v, w$ integers. Deduce that $1, \alpha, \beta$ form an integral basis for $K$.