Suppose I have a field $K=\mathbb{Q}(\sqrt{-d})$.
How does one describe it's integral ideals?
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What do you wanna know exactely? There is an exact list of PIDS: https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one. As a general corollary of Unique factoritazion Theorem every Ideal can be generated by 2 elements: $I = (a,b)$. – Maffred Mar 10 '16 at 17:44
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I've been given a specific field and am required to describe all the integral ideals of K up to what I assume is the number just below the Minkowski bound – user321680 Mar 10 '16 at 17:56
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You have to check every prime number under the minkosky bound and prove or disprove $pO_K$ is principal or not. – Maffred Mar 10 '16 at 18:01
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Could you possibly run me through an example for an arbitrary, simple d? – user321680 Mar 10 '16 at 18:11
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There are a lots of examples on this site. Start for example from here: http://math.stackexchange.com/questions/1313030/prove-the-class-number-of-mathbbz-sqrt-5-is-2 – Maffred Mar 10 '16 at 18:43