Re: [sage-support] Calculate discrimanent of relative number fields

[David Loeffler]

On 29 August 2012 12:54, Cindy <cindy42...@gmail.com> wrote:
> Hi,
>
> Given a cyclotomic field Q(zeta_n), where zeta_n is a primitive nth root of
> unity, with maximal real subfield F, how can I calculate the discriminant of
> K/F?

You need to use the "relativize" command to create the field extension
K / F. Here's an example for the 13th cyclotomic field:

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| Sage Version 5.2, Release Date: 2012-07-25 |
| Type "notebook()" for the browser-based notebook interface. |
| Type "help()" for help. |
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sage: K.<zeta> = CyclotomicField(13)
sage: Krel = K.relativize(zeta + zeta^(-1), "w")
sage: Krel
Number Field in w0 with defining polynomial x^2 - w1*x + 1 over its base field
sage: Krel.base_field()
Number Field in w1 with defining polynomial x^6 + x^5 - 5*x^4 - 4*x^3
+ 6*x^2 + 3*x - 1

# so now Krel is QQ(zeta13) as an extension of F = QQ(zeta13 + zeta13^(-1))

sage: Krel.relative_discriminant()
Fractional ideal (w1^5 - 5*w1^3 + 4*w1)
# (an ideal of F)

sage: Krel.relative_different()
Fractional ideal ((w1^3 - 2*w1)*w0 - w1^2)
# (an ideal of Krel)

Regards, David

[Cindy]

Thanks a lot David! It works ^^

Cindy