How to compute the different ideal of a number field extension in Sage?

Ys1123
15 ●7

Let $K/F$ be $\mathbb{Q}(\sqrt[3]{2},\zeta_3)/\mathbb{Q}(\sqrt[3]{2})$ . I want to compute the different ideal $\mathfrak{d}$ for $K/F$ and check the formula of $N_{K/F} (\mathfrak{d})=\Delta_{K/F}$ where $\zeta_n$ is a $n$ -th root of unity and $\Delta_{K/F}$ is the relative discriminat of $K/F$ .

But, at first, I tried to compute one more easier case for $\mathbb{Q}(\sqrt{-5})/\mathbb{Q}$ . It might be $\mathfrak{d}_{\mathbb{Q}(\sqrt{-5})/\mathbb{Q}}=(2\sqrt{-5})$ since $\Delta_{\mathbb{Q}(\sqrt{-5})/\mathbb{Q}}=(20)$ .

My attempt is following:

x=polygen(ZZ, 'x')
K.<a>=NumberField(x^2+5)
gp(K.diff)

But this makes an error. Could you show me the correct code for this and for original setting?

Comments

In Sage/Python, you need to add the parenthesis to "call" the function. Without the parenthesis, it returns the function itself:

sage: K.discriminant
<bound method NumberField_quadratic.discriminant of Number Field in a with defining polynomial x^2 + 5>
sage: K.discriminant()
-20

Also the function (also called method) diff does not exists for K, that is, writting K.d and hitting the TAB key does not list any method called diff for the number field K.

Sébastien ( 0 years ago )

OK, then how can I get the different ideal (not the discriminant)?

Ys1123 ( 0 years ago )

If you want to list methods of K containing the word "ideal", you may do:

sage: K.*ideal*?
K.fractional_ideal
K.ideal
K.ideal_monoid
K.idealchinese
K.ideals_of_bdd_norm
K.principal_ideal
K.unit_ideal
K.zero_ideal

If you want to list methods of K containing the word "different", you may do:

sage: K.*different*?
K.absolute_different
K.different
K.relative_different

Sébastien ( 0 years ago )

add a comment

sage: x = polygen(ZZ, 'x') sage: K.<a> = NumberField(x^3-2) sage: PK.<t> = K[] sage: L.<zeta3> = K.extension(t^2+t+1) sage: L.relative_different() Fractional ideal ((1/3*a^2 + 2/3*a + 1/3)*zeta3 + 2/3*a^2 + 1/3*a - 1/3)

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How to compute the different ideal of a number field extension in Sage?

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