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I am asking for clarification on the answer of Hagen von Eitzen to the question "Prove that the class number of $\mathbb{Z}[\sqrt{-5}]$ is $2$.

He claims that given any non-principal ideal $J$ in $\mathbb{Z}[\sqrt{-5}]$ the ideal product $(2, 1 + \sqrt{-5}) * J$ is principal. This, together with the fact that $(2, 1 + \sqrt{-5})$ is non-principal of course suffices to establish that the ideal class group of $\mathbb{Z}[\sqrt{-5}]$ has exactly two elements, hence giving the proof I am looking for. I however to not see how to show this using his hint of considering elements of minimal norm.

As Hagen von Eitzen has not answered a question for further elaboration on the original question, I hope to get some interaction by asking a new question. If this is not appropriate, please close or delete this question.

Sven-Ole Behrend
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  • Have you looked at the treatment of imaginary quadratic fields in Mike Artin's Algebra? He has a section on ideal class groups, and I think he gives an elementary geometric argument for this example having class number $2$ (i.e., all nonprincipal ideals are equal up to scaling factors). – KCd Feb 08 '22 at 23:40

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For any non-zero ideal $I$ there is an element $a\in I-0$ such that $N(a)\le 10 N(I)$. So the inverse of $I$ in the class group is represented by an ideal of norm $\le 10$. Enumerating all those ideals of norm $\le 10$ you'll find that they are either principal or in the class of $(2,1+\sqrt{-5})$ (enumerating the prime ideals of norm $\le 10$ is enough).

For the construction of $a$: let $m=\lfloor \sqrt{N(I)}\rfloor+1 $ and consider the set $S=\{u+v\sqrt{-5}, u,v \in 1\ldots m\}$. We have listed $m^2>N(I)$ elements of $\Bbb{Z}[\sqrt{-5}]$ so there must be two elements of $S$ that are the same in $\Bbb{Z}[\sqrt{-5}]/I$ ie. $(u_1-u_2)+(v_1-v_2)\sqrt{-5}\in I-0$ with $|u_1-u_2|\le m,|v_1-v_2|\le m$.

reuns
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