View Single Post 2018-01-13, 18:07   #3
carpetpool

"Sam"
Nov 2016

22×83 Posts Quote:
 Originally Posted by Nick 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
Yes, Nick.

https://en.wikipedia.org/wiki/Ideal_class_group

I am not sure if this grasps the same concept I addressed:

Quote:
 Originally Posted by Wikipedia; In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K.
Quote:
 Originally Posted by Wikipedia; If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that (a)I = (b)J. (Here the notation (a) means the principal ideal of R consisting of all the multiples of a.)
According to the second paragraph, for some number field K, and Ok is its ring of integers, we have a "sub-ring" of integers Ok/p such that an ideal of norm p divides every integer in the "sub-ring". We can ask which ideals Q multiplied by the ideal P (of norm p), will make PQ a principal ideal?

I.e. The Question I was asking is --- how many different, hence distinct "sub-rings" are contained in the field K?  