Consider the field extension $\mathbb Q(\alpha)/\mathbb Q$ obtained by adjoining a root of the polynomial $f(x)=x^3+10x+1$. The discriminant is $-4023$, a prime number, and the Minkowski bound tells us that in order to study the class ideal group, we only need to consider (prime) ideals of order up to $17$.
Q1) In class, we approached this problem by considering prime factorizations of all principal ideals of the form $(\alpha-a)$ for $-8\le a \le 8$. How do we know that all prime ideals of order $\le 17$ must appear in the factorization of these ideals?
Q2) In the factorization of $(\alpha-1)$, we used $N(\alpha-1) = 2\times2\times3$. Why does it follow from here that $(\alpha-1) = P_2^2P_3$, where the subscript indicates the norm of the ideal? How do we know that the two ideals of norm $2$ are the same? And how do we know that it doesn't factor like $P_4P_3$? After all, a prime ideal might have every possible prime power norm, right? (at least, if we are considering a finite field extension over $\Bbb Q$)