$\mathbb{A} \cap \mathbb{Q}[\alpha]$ is a principal ideal domain with $\alpha^3=\alpha+7$

Question

I'm trying to solve exercise 20 of chapter 5 of the book Number Fields by Marcus. $$ \text{Prove that $\mathbb{A} \cap \mathbb{Q}[\alpha]$ is a principal ideal domain when $\alpha^3 = \alpha+ 7$} $$ I just solved the exercises 15 to 19 of the same chapter using Minkowski's bound and studying the factorization of $2R$ and $3R$.
This time when I apply Minkowsi's bound i get that $||J|| \le 10.27$ and I have to study the factorization of $2R,3R,5R$ and $7R$.
I know that $\operatorname{disc}(\alpha)=\operatorname{disc}(R)=-1319$ that is a prime number so I know that $2R,3R,5R$ and $7R$ are unramified because they do not divide the discriminant but I can't say anything else.
In the book is contained the hint that $7 = \alpha^3- \alpha$ but I don't know how to use it.