Let $K=\mathbb{Q}(\alpha)$, where $\alpha^{3}=5 \alpha-8$

(a) Show that $[K: \mathbb{Q}]=3$.

(b) Let $\beta=\left(\alpha+\alpha^{2}\right) / 2$. By considering the matrix of $\beta$ acting on $K$ by multiplication, or otherwise, show that $\beta$ is an algebraic integer, and that $(1, \alpha, \beta)$ is a $\mathbb{Z}$-basis for $\mathcal{O}_{K} \cdot$ [The discriminant of $T^{3}-5 T+8$ is $-4 \cdot 307$, and 307 is prime.]

(c) Compute the prime factorisation of the ideal (3) in $\mathcal{O}_{K}$. Is (2) a prime ideal of $\mathcal{O}_{K} ?$ Justify your answer.