Proving the class number of $\mathbb{Z}[\sqrt{-5}]$ is 2 via ideal equivalence

Asked May 18 '25 at 13:29

Active May 19 '25 at 12:25

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I want to prove that the class number of $A:=\mathbb{Z}[\sqrt{-5}]$ is $2$. I want to show that every non principal ideal $J$ of $A$ is equivalent to $I:=2A+(\sqrt{-5}-1)A$. Then since. I showed that $I$ is non principal. I tried to consider an elemnt $x$ of minimal nonzero norm in $J$ and tried to express $J$ with $x$ and $I$, but I haven't succeeded so far.

Thanks for your help!

edited May 19 '25 at 12:25

asked May 18 '25 at 13:29

Lasky

4

I disagree with the closing of this question. It should be re open. – Enguerrand Moulinier May 18 '25 at 15:09
This has been solved here already, see for example here. For different methods, see also here, or here, and many more posts. – Dietrich Burde May 21 '25 at 11:55

Proving the class number of $\mathbb{Z}[\sqrt{-5}]$ is 2 via ideal equivalence

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