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I'm trying to understand imaginary quadratic fields and the ideal class group seems to be really important, yet I cannot find a simple explanation of it and why it's important. I understand that it is related to unique factorization, yet I don't know why. If someone could give a simple explanation of it and maybe a simple example of it I'd really appreciate it.

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  • I can recommend a new, and inexpensive, book by J. L. Lehman, called Quadratic Number Theory. Does this in expanded, form. – Will Jagy Mar 31 '19 at 20:10
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    https://bookstore.ams.org/dol-52/ – Will Jagy Mar 31 '19 at 20:12
  • The class group of $\Bbb{Z}[\sqrt{-5}]$ has two elements $(1),(2,1+\sqrt{-5})$. Any ideal $I$ of $\Bbb{Z}[\sqrt{-5}]$ is of the form $I=(a)$ for some $a \in \Bbb{Z}[\sqrt{-5}]$ or $(2a,(1+\sqrt{-5})a)$ for some $a \in (2,1+\sqrt{-5})^{-1} \iff 2a \in (2,1+\sqrt{-5})$. And $(2,1+\sqrt{-5}) = { 2b+(1+\sqrt{-5})c, b,c\in \Bbb{Z}[\sqrt{-5}]}$. – reuns Mar 31 '19 at 20:17

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