Quote:
Originally Posted by carpetpool
K is a number field,
h is its class number > 1,
P and Q are nonprincipal prime ideals in K, so are P_{n} and Q_{n},
[G, G_{2}, G_{3},... G_{n}] (ideal groupings) are the groupings of all nonprincipal prime ideals such that the product of any two prime ideals P and Q in the same group G_{n} 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:
http://www.mersenneforum.org/showthread.php?t=21877