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Old 2018-01-13, 05:09   #1
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"Sam"
Nov 2016

23·41 Posts
Post Ideal groupings in number fields

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

The number of groupings Gn is not necessarily the same as its class number.
However, the maximum number of groupings Gn 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, G2]. 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 G2.

(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, G2, and G3]

If d = 1 or 7, then P belongs in Group G.
If d = 2 or 6, then P belongs in Group G2.
If d = 3 or 5, then P belongs in Group G3.

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, G2, G3,... Gn] 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 Gn for each of the prime cyclotomic fields, as well as any other useful information.
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