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Old 2018-01-13, 18:13   #4
carpetpool
 
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"Sam"
Nov 2016

23·41 Posts
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Quote:
Originally Posted by Nick View Post
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:
http://www.mersenneforum.org/showthread.php?t=21877
I didn't notice your question there, and yes this is true (I don't have a proof although I'm sure there is one someone already discovered).

Example:

K= Q(sqrt(-5)), the ideals P = <x-2,3>, Q = <x-3,7>, and R = <x-8,23> are non-principal.

It follows that PQ and PR are principal ideals, so QR is also a principal ideal.
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