I am trying to define the polynomial ring $\Bbb{Z}[j]$ where $j$ is a primitive 3rd root of unity, i.e. $j = e^{\frac{2}{3} \pi i} \in \Bbb C$.
I tried this:
P.<x> = PolynomialRing(QQ)
j = exp((2/3)*pi*i)
P[j]
But I get the error:
NotImplementedError: ring extension with prescribed embedding is not implemented
The ring you need is the ring of Eisenstein integers.
It is implemented in Sage as EisensteinIntegers().
Try this:
sage: P.<j> = EisensteinIntegers()
sage: P
Eisenstein Integers in Number Field in j with defining polynomial x^2 + x + 1 with j = -0.50000000000000000? + 0.866025403784439?*I
sage: j
j
sage: j^2
-j - 1
sage: j^3
1
sage: 1 + j + j^2
0
Asked: 2021-12-06 15:51:03 +0100
Seen: 248 times
Last updated: Dec 06 '21
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