kummer extension on SAGE

Question

I want to calculate the relative discriminant of field extensions of this kind: $$\mathbb{Q}(\zeta_5)(\sqrt[5]{a})$$ Where $a \in \mathbb{Q}(\zeta_5)$. So I use SAGE and make this calculations:

K.<b>=CyclotomicField(5);    //my field base
alpha=1+3*b^2;               //an element of my field base
f=(1+3*b^2).minpoly();       //its minimal polynomial
f.is_irreducible()           //is it irreducible?
R.<a>=K.extension(f)         //the field extension of my field base
R.relative_discriminant()    //the calculation of the relative discriminant

But when I execute it, appears this error

defining polynomial (x^4 - x^3 + 6*x^2 + 14*x + 61) must be irreducible

But it is irreducible, what am I doing wrong? Or how can I solve this?

Answer 1

I'm not sure whether this is the best place to ask sage questions, but the problem is here that the minimal polynomial $f$ of $\alpha$ is of course not irreducible over $K$. Probably you meant to do the following:

K.<b>=CyclotomicField(5)
alpha=1+3*b^2
P.<x>=K[]
R.<a>=K.extension(x^5-alpha)
R.relative_discriminant()  

which gives:

Fractional ideal (-203125*b^3 - 121875*b^2 - 3125*b - 50000)