In this question, I asked about some non-cyclic and hand-doable example of ideal class group, and Yong Hao Ng the example with minimal discriminant is $K = \mathbb{Q}(\alpha)$ with $\alpha^{3} + 11\alpha + 21 = 0$. With help of SAGE and some tedious computations, I proved that the class group is generated by two ideal classes $[(3, \alpha-1)]$ and $[(3, \alpha)]$ and both have order 2.
To show that $(3, \alpha – 1)$ is not a principal ideal, I proved that $\epsilon = \alpha^{2}- \alpha – 4$ is a fundamental unit of $K$ by using Artin's inequality. Then if $(3, \alpha – 1)=(\beta)$ for some $\beta\in \mathcal{O}_{K} = \mathbb{Z}[\alpha]$, $(\alpha+2) = (3, \alpha – 1)^{2} = (\beta^{2})$ implies that $v(\alpha + 2) = \beta^{2}$ for some unit $v\in U_{K} =\mathcal{O}_{K}^{\times}$.
We can assume $v = 1$ or $v = \epsilon$, and for example, if $\alpha + 2 = \beta^{2} = (a\alpha^{2} + b\alpha +c)^{2}$, then by expanding this, we should have $c^{2} + 42ab = 2$, which is impossible by viewing the equation mod 3. Similarly, we can show that there's no $\beta\in \mathbb{Z}[\alpha]$ s.t. $\beta^{2} = \epsilon(\alpha + 2)= \alpha^{2}- 17\alpha – 29$ (by mod 7).
I can also prove that $(3, \alpha)$ is not a principal ideal since $(3, \alpha)(13, \alpha + 3) = (\alpha + 3)$ and $(13, \alpha + 3)$ is not a principal ideal by the similar method.
To prove that the class group is isomorphic to Klein 4-group, the only thing we have to do now is $[(3, \alpha -1)] \neq [(3, \alpha)]$, which is equivalent to that $\mathfrak{p}_{3}\mathfrak{p}_{3}' = (3, \alpha -1)(3, \alpha)$ is not a principal ideal.
However, I can't prove this by the above method, since I can't find any contradiction.
We have $(\mathfrak{p}_{3}\mathfrak{p}_{3}')^{2} = (-\alpha-1)$, so we only need to show that $(-\alpha – 1)v =\beta^{2}$ is not possible for any $v\in U_{K}$ and $\beta\in \mathcal{O}_{K}$.
As before, $-\alpha – 1 = \beta^{2}$ doesn't have a solution by observing the coefficient of $\beta^{2}$. (-1 is not a square mod 3).
However, I can' prove that $(\alpha^{2} – \alpha – 4)(-\alpha – 1) = 16\alpha + 25$ is not a square in $\mathcal{O}_{K}$. This is equivalent to show that the following system of Diophantine equation
$$
b^{2} + 2ac – 11a^{2} = 0 \\
2bc + 22ab – 21a^{2} = 16\\
c^{2} + 42ab = 25
$$
doesn't have any integer solution $(a, b, c)$. Clealy, the last equation doesn't give any contradiction modulo any prime $p$.
So I try to find other nontrivial relation among ideal classes using SAGE, but I failed to find any one of them that help me to prove that the above ideal is not principal. Also, I tried $v = \epsilon^{2k+1}$, instead of $\epsilon$, and nothing gives a contradiction.
Also, it is enough to show that one of the following (prime) ideals
$$
\mathfrak{p}_{3}'' = (3, \alpha + 1) \\
\mathfrak{p}_{17} = (17, \alpha – 2) \\
\mathfrak{p}_{23} = (23, \alpha – 8) \\
\mathfrak{p}_{29} = (29, \alpha + 4) \\
\mathfrak{p}_{29} = (29, \alpha – 10)
$$
are not principal. But I failed for every above ideal. Do you have any idea? Thanks in advance.
Edit: I just find a way that I can't do without SAGE. If $\sqrt{16\alpha+25} = \beta\in K$, then $\beta$
$$
\left(\frac{\beta^{2}-25}{16}\right)^{3} + 11\left(\frac{\beta^{2} – 25}{16}\right) + 21 = \frac{\beta^{6} – 75\beta^{4} + 4691\beta^{2} – 9}{4096} = 0.
$$
However, according to SAGE, the polynomial $x^{6} – 75x^{4} + 4691x^{2} – 9$ is irreducible over $\mathbb{Q}$, so the degree of $\beta$ is 6, which contradicts to $\beta\in \mathbb{Q}(\alpha)$. However, I think I can't show that the degree 6 polynomial is irreducible (and even it is hard to compute it by hands.)
Best Answer
As you observed, $(\mathfrak{p}_{3}\mathfrak{p}_{3}')^{2} = (\alpha+1)$, so to prove $\mathfrak{p}_{3}\mathfrak{p}_{3}'$ is not principal, it suffices to show none of $\pm(\alpha+1)$ and $\pm \epsilon(\alpha+1)$ is a square in $\mathcal{O}_K$, where $\epsilon = \alpha^{2}- \alpha - 4$ is the fundamental unit you identified. There is a very efficient way to do this:
Let $f(x)=x^3+11x+21$, then $101\mid f(6)$. Since the ring of integer is monogenic, we have a homomorphism: $$\mathcal{O}_K \to \mathbb{F}_{101} \qquad \alpha \mapsto 6$$ under this, $\pm(\alpha+1)$ is mapped to $\pm 7$, but it is easy to check $\pm 7$ is not a quadratic residue modulo $101$. Hence $\pm(\alpha+1)$ is not a square in $\mathcal{O}_K$. The same prime $101$ does not work for $\pm \epsilon(\alpha+1)$, because they are mapped to $\pm (6^2-6-4)(6+1)$, both are quadratic residue of $101$.
Therefore we try another prime. Note that $29\mid f(10)$, use the homomorphism: $\mathcal{O}_K \to \mathbb{F}_{29}, \alpha \mapsto 10$, $\pm \epsilon(\alpha+1)$ is mapped to $\pm (10 + 1) (10^2 - 10 - 4)$ which are quadratic nonresidue modulo $29$.