quotient of algebraic integer rings

asked 2019-11-04 06:39:09 +0200

This is my first time using sage so this might be a stupid question:

I want to construct the field $K=\mathbb{Q}(\sqrt{2}, \frac{-1+\sqrt{3}i}{2})=\mathbb{Q}(\alpha)$, where $\alpha$ is a primitive element. Denoting its ring of algebraic integers $\mathcal{O}_K$, I want to compute the quotient ring $\mathcal{O}_K/\mathbb{Z}[\alpha]$.

My code is like:

K.<d> = QQ.extension(x^2-2)

L.<w> = K.extension(x^2+x+1)

a = L.primitive_element()

O = L.ring_of_integers()

za = ZZ[a]

print O.quotient(za)

But I always get an error: TypeError: unable to convert Relative Order in Number Field in w0 with defining polynomial x^2 + (2d + 1)x + d + 3 over its base field to Number Field in w with defining polynomial x^2 + x + 1 over its base field

Any help is appreciated. Thanks.

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I'm not sure how to fix the type error. But the ring $\mathbb{Z}[\alpha]$ is not an ideal of $\mathcal{O}_K$. But it is an (additive) subgroup. You can try to write $\alpha^j = \sum a_{ij} \alpha_i$ where $ { \alpha_i } $ is an integral basis for $\mathcal{O}_K$. Then you can compute the smith normal form of $[a_{ij}]$ and that should tell you the structure of the abelian group $\mathcal{O}_K/\mathbb{Z}[\alpha]$.

David Tweedle gravatar imageDavid Tweedle ( 2019-11-08 14:14:34 +0200 )edit