How to prove that pZ is a maximal ideal for the ring of integers?

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morphism
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There probably are analogues of the genus and canonical divisor for number fields. I don't know what they could be though. Someone more well-versed in Arakelov geometry would be able to answer.

At least I can tell you that the class group of a quadratic field can indeed be arbitrarily large: Gauss conjectured, and Heilbronn proved, that the class number of the imaginary quadratic field ##\mathbb{Q}(\sqrt{-d})## tends to infinity as ##d \to \infty##.
 

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