 Solution

Dedekind's criterion factors $(p)$ according to the factorization of $X^3-7$ modulo $p$ (the monogenic assumption removes the index obstruction). With $\alpha =\sqrt[3]{7}$,Minkowski's bound is below 11, so classes are generated by prime ideals over $2,3,5,7$. Relations from the displayed factorizations and principal ideals reduce them to $\mathfrak{p}=(3,\alpha -1)$ with ${\mathfrak{p}}^3$ principal. Part (a) shows $\mathfrak{p}$ is not principal, so $\mathrm{Cl}(K)\simeq \mathbb{Z}/3\mathbb{Z}$.