I have the following exercise:
The cubic field with the smallest discriminant, in absolute value, is $\mathbb{Q}(\alpha)$ with $\alpha$ a root of $T^3-T+1$ and with ring of integers $\mathbb{Z}[\alpha]$.
I want to list all the ideals of norm less or equal to $10$.
What I already have:
Minkowski bound is 1,065.. or 1.3... depending on whether $r_2$ is 0 or 1. Hence the class group is the trivial one.
The discriminant of the number field is $-23$ as it is the discriminant of the given polynomial $T^3-T+1$
The smallest value the absolute discriminant of a cubic number field can take is $23$. It is given by $\mathbb{Q}[x]/(x^3-x^2+1)$, whose discriminant is $-23$. We can use Minkowski's bound and the Stickelberger relation for this exercise. Is the listing of ideals part of the exercise? This is not written there, it seems.