To check whether a given ideal of an order is principal or not

asked 2016-02-11 08:53:14 -0600

user2893829 gravatar image

Suppose $\theta$ is a root of a irreducible monic polynomial $f$ of degree $n$. (In practice, I would like to deal with $n=3$ case.) Then, define the ideal class group of $\mathbb{Z}[\theta]$, $C(\mathbb{Z}[\theta])$ by the set of invertible fractional ideals modulo principal ideals.

Given a polynomial $f$ and a fractional ideal $I$ of $\mathbb{Z}[\theta]$, is there any way to decide I is principal or not?

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The reference manual gives the solution (if I understand you question correctly):

sage: K.<a, b> = NumberField([x^2 - 23, x^2 + 1])
sage: I = K.ideal([7, (-1/2*b - 3/2)*a + 3/2*b + 9/2])
sage: I.is_principal()
sage: I
Fractional ideal ((1/2*b + 1/2)*a - 3/2*b - 3/2)
B r u n o gravatar imageB r u n o ( 2016-04-02 12:04:53 -0600 )edit