On 29 August 2012 12:54, Cindy <[email protected]> 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: ---------------------------------------------------------------------- | Sage Version 5.2, Release Date: 2012-07-25 | | Type "notebook()" for the browser-based notebook interface. | | Type "help()" for help. | ---------------------------------------------------------------------- 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 -- You received this message because you are subscribed to the Google Groups "sage-support" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-support?hl=en.
