How to compute an ideal of specific norm?

number_theory

asked 2021-07-16 09:01:52 +0200

prathamlalwani
33 ●1 ●5

updated 2021-07-27 10:17:41 +0200

FrédéricC

5125 ●3 ●43 ●111

So I was wondering if there was a way to compute ideals of a certain norm something like:

sage: K = QuadraticField(2) 
sage: K.ideal_of_norm(2)
[Fractional ideal (a)]

something like this.

edit retag flag offensive close merge delete

Comments

This related question might help:

Ask Sage question 51803: Prime ideals up to a specific norm

slelievre ( 2021-07-18 10:48:10 +0200 )edit

add a comment

1 Answer

Sort by » oldest newest most voted

answered 2021-07-20 06:10:38 +0200

nbruin
4143 ●3 ●45 ●90

Your question can be solved using ideal factorization. If the norm you're looking for does not have extremely large prime factors in it, it's probably easiest to construct your ideal by multiplying ideals together of appropriate prime power norms. Hence, you'd want to find ideals of prime power norm, for given prime p:

sage: K.<a>=QuadraticField(2)
sage: OK=K.maximal_order()
sage: (5*OK).factor()
Fractional ideal (5)
sage: (7*OK).factor()
(Fractional ideal (-2*a + 1)) * (Fractional ideal (2*a + 1))

As you can see, 5 is inert, so norms of ideals can only have an even power of 5 in them. 7 on the other hand, splits, so you have two different choices for ideals of norm 7. You can multiply them together in all kinds of ways to make ideals with norm any power of 7.

edit flag offensive delete link

add a comment

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2021-07-16 09:01:52 +0200

Seen: 410 times

Last updated: Jul 20 '21

How to compute an ideal of specific norm? edit