Finding a certain ideal

asked 2024-02-12 08:37:27 +0200

imbluedabedee gravatar image

updated 2024-02-12 21:30:19 +0200

Hey SAGE-Community,

I am new to SAGE and looking for some help for a small university project. Given 𝒪K as the ring of integers to K=ℚ(√5). Given also a totally negative number d ∈ 𝒪K2. How can I calculate the largest integral ideal f in K whose square divides d such that d is a square modulo 4 f?

I guess that both questions are rather easily done in SAGE but I am missing the necessary experience. Anyways, thanks a lot in advance!

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Is this an example of what you want? K.<a> = NumberField(x^2 - 5); OK = K.ring_of_integers(); d = -4; f = OK.fractional_ideal(2); I = 4*f; I.reduce(d) in set(I.reduce(c^2) for c in I.residues()) gives True. It could help someone write an answer.

rburing gravatar imagerburing ( 2024-02-12 11:47:27 +0200 )edit

Hey rburing, thanks for your feedback. This sounds correct, but I am still not sure on how to find the ideal in general.

imbluedabedee gravatar imageimbluedabedee ( 2024-02-13 15:11:56 +0200 )edit