Picard group of $\mathbb Z[\sqrt{-5}]$

Question

I search for a simple proof for the fact that $\operatorname{Pic}(\mathbb Z[\sqrt{-5}])=\mathbb Z/2\mathbb Z$, where $\operatorname{Pic}(R)$ is the Picard group of the ring $R$ - the set of isomorphism classes of f.g. projective $R$-modules of of rank $1$.

I appreciate anybody cooperating!

user26857 · Accepted Answer · 2014-12-27T17:45:47.727

4

For Dedekind domains we have $\operatorname{Pic}(R)\simeq\operatorname{Cl}(R)$, where $\operatorname{Cl}(R)$ is the ideal class group of $R$. Your case is treated here in detail.

edited Dec 27 '14 at 17:45

answered Dec 14 '14 at 16:22

user26857

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For a more advanced approach using the Minkowski's bound you can have a look at these lecture notes. – user26857 Dec 14 '14 at 16:26

Picard group of $\mathbb Z[\sqrt{-5}]$

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