I am trying to define the polynomial ring Z[j] where j is a primitive 3rd root of unity, i.e. j=e23πi∈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 implementedThe 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
0Thanks! Is it possible to express Z[j](2−5j) as well? I tried P.quotient((2-5*j)), but I am getting IndexError: the number of names must equal the number of generators.
What is Z[j]2−5j ?
Asked: 3 years ago
Seen: 263 times
Last updated: Dec 06 '21
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