quotient of algebraic integer rings
This is my first time using sage so this might be a stupid question:
I want to construct the field K=Q(√2,−1+√3i2)=Q(α), where α is a primitive element. Denoting its ring of algebraic integers OK, I want to compute the quotient ring OK/Z[α].
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.
I'm not sure how to fix the type error. But the ring Z[α] is not an ideal of OK. But it is an (additive) subgroup. You can try to write αj=∑aijαi where αi is an integral basis for OK. Then you can compute the smith normal form of [aij] and that should tell you the structure of the abelian group OK/Z[α].