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-   -   Ideal groupings in number fields (https://www.mersenneforum.org/showthread.php?t=22922)

carpetpool 2018-01-13 05:09

Ideal groupings in number fields
 
K is a number field,
h is its class number > 1,
P and Q are [B]non-principal[/B] prime ideals in K, so are P[SUB]n[/SUB] and Q[SUB]n[/SUB],
[G, G[SUB]2[/SUB], G[SUB]3[/SUB],... G[SUB]n[/SUB]] (ideal groupings) are the groupings of all [B]non-principal[/B] prime ideals such that the product of any two prime ideals P and Q in the same group G[SUB]n[/SUB] is principal.
d is the exponent on the class group generator of any prime ideal.

The number of groupings G[SUB]n[/SUB] is not necessarily the same as its class number.
However, the maximum number of groupings G[SUB]n[/SUB] 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[SUB]2[/SUB]]. 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[SUB]2[/SUB].

(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[SUB]2[/SUB], and G[SUB]3[/SUB]]

If d = 1 or 7, then P belongs in Group G.
If d = 2 or 6, then P belongs in Group G[SUB]2[/SUB].
If d = 3 or 5, then P belongs in Group G[SUB]3[/SUB].

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[SUB]2[/SUB], G[SUB]3[/SUB],... G[SUB]n[/SUB]] 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[SUB]n[/SUB] for each of the prime cyclotomic fields, as well as any other [I]useful[/I] information. :smile:

Nick 2018-01-13 10:24

[QUOTE=carpetpool;477423]K is a number field,
h is its class number > 1,
P and Q are [B]non-principal[/B] prime ideals in K, so are P[SUB]n[/SUB] and Q[SUB]n[/SUB],
[G, G[SUB]2[/SUB], G[SUB]3[/SUB],... G[SUB]n[/SUB]] (ideal groupings) are the groupings of all [B]non-principal[/B] prime ideals such that the product of any two prime ideals P and Q in the same group G[SUB]n[/SUB] is principal. [/QUOTE]
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:
[URL]http://www.mersenneforum.org/showthread.php?t=21877[/URL]

carpetpool 2018-01-13 18:07

[QUOTE=Nick;477432]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:
[URL]http://www.mersenneforum.org/showthread.php?t=21877[/URL][/QUOTE]

Yes, Nick.

[url]https://en.wikipedia.org/wiki/Ideal_class_group[/url]

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

[QUOTE=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]

[QUOTE=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.)[/QUOTE]

According to the second paragraph, for some number field K, and O[SUB]k[/SUB] is its ring of integers, we have a "sub-ring" of integers O[SUB]k[/SUB]/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?

carpetpool 2018-01-13 18:13

[QUOTE=Nick;477432]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?
[B]For example, if PQ and QR are principal, does it follow that PR is? What about PP?[/B]

More generally, if you are interested in class groups, it would help to learn a little group theory. You could start here:
[URL]http://www.mersenneforum.org/showthread.php?t=21877[/URL][/QUOTE]

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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