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Old 2018-01-13, 10:24   #2
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Dec 2012
The Netherlands

5·353 Posts

Originally Posted by carpetpool View Post
K is a number field,
h is its class number > 1,
P and Q are non-principal prime ideals in K, so are Pn and Qn,
[G, G2, G3,... Gn] (ideal groupings) are the groupings of all non-principal prime ideals such that the product of any two prime ideals P and Q in the same group Gn is principal.
A field has only 2 ideals, both principal. Do you mean prime ideals of some subring of K such as its ring of integers?
You appear to be assuming you have an equivalence relation here. Is that really so?
For example, if PQ and QR are principal, does it follow that PR is? What about PP?

More generally, if you are interested in class groups, it would help to learn a little group theory. You could start here:
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