K is a number field,

h is its class number > 1,

P and Q are

**non-principal** 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

**non-principal** prime ideals such that the product of any two prime ideals P and Q in the same group G

_{n} is principal.

d is the exponent on the class group generator of any prime ideal.

The number of groupings G

_{n} is not necessarily the same as its class number.

However, the maximum number of groupings G

_{n} is h-1.

Let's take a look at some examples:

Lemma I: If the only ideal grouping in K are G, then the product of any two non-principal ideals is principal. The exponent on the class group generator of P is 1.

For K=Q(sqrt(-5)), h = 2, and the groupings in K are [G].

P and Q must be in this group and d = 1. Since there is only ideal grouping G, this implies that the product of any two non-principal ideals are principal (a restate of Lemma I).

In fact, this is true for all fields K with class number 2, and some other fields with class number h > 2.

K=Q(sqrt(-23)) has class number h = 3, and there is also one ideal grouping G, hence Lemma I is true here.

K=Q(sqrt(-47)) has class number h = 5, however Lemma I is not true. There are two ideal groupings [G, G

_{2}]. One can determine which group P belongs in.

If d = 1 or 4, then P belongs in Group G.

If d = 2 or 3, then P belongs in Group G

_{2}.

(P is principal otherwise)

One common conclusion to come to is if K is a number field with class number h, then the number of ideal groupings in K divides h-1. The short and easy answer to this is no, this is not always true. (It is sometimes.)

The field K = Q(sqrt(-95)) has class number h = 8. The groupings are [G, G

_{2}, and G

_{3}]

If d = 1 or 7, then P belongs in Group G.

If d = 2 or 6, then P belongs in Group G

_{2}.

If d = 3 or 5, then P belongs in Group G

_{3}.

This field is interesting because the number of ideal groupings (3) does not divide h-1 (7), yet the distribution of prime ideals in these groups are equal. It is obvious that no prime ideal will have exponent d = 4 on its class group generator.

For quadratic fields, it seems pretty easy to work out. What about for the nth cyclotomic fields Kn, for prime n?

Excluding K2-K19 (because h = 1), we have g = [G, G

_{2}, G

_{3},... G

_{n}] the number of ideal groupings (as I have first defined) in Kn, and (n,g):

(23,1)

(29,1)

(31,2)

(37,1)

(41,3)

(43,5)

(47,15)

I have also computed the series of exponents d in each of the following ideal groupings for Kn. (Private Message me if you want any of these references.) I spend most of my PC power currently trying to classify the ideal groupings for larger cyclotomic fields.

It would be nice to know if there is already a list of the number of ideal groupings G

_{n} for each of the prime cyclotomic fields, as well as any other

*useful* information.