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Trying to find prime factorization of ideals in number fields

asked 8 years ago

nebuckandazzer gravatar image

updated 8 years ago

Let L=Q(5,i) and K=Q(5). Let OK and OL be the rings of algebraic integers of K and L. It can be checked that

2OK=2,5+12

I want to find the factorization of the ideal 2,5+1OL in OL ?

The problem I am having is this. I don't know the syntax for the ideal generated by 2,5+1OL.

What to do ?

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answered 8 years ago

slelievre gravatar image

updated 8 years ago

Is this what you are looking for?

Define the number fields and their rings of integers.

sage: K.<a> = NumberField(x^2+5)
sage: L.<i> = K.extension(x^2+1)
sage: OK = K.ring_of_integers()
sage: OL = L.ring_of_integers()

Define the ideals generated by 2 and by a+1.

sage: J = OL.principal_ideal(2)
sage: JJ = OL.principal_ideal(a + 1)

Take the sum.

sage: H = J + JJ; H
Fractional ideal (-i + 1)

Prime factors.

sage: H.prime_factors()
[Fractional ideal (-i + 1)]

sage: H.is_prime()
True
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Asked: 8 years ago

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Last updated: Nov 15 '16