# Polynomial ring with complex elements.

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

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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

more

Thanks! Is it possible to express $\frac{\Bbb{Z}[j]}{(2-5j)}$ as well? I tried P.quotient((2-5*j)), but I am getting IndexError: the number of names must equal the number of generators.

( 2021-12-06 17:54:43 +0200 )edit

What is $\frac{\mathbb Z[j]}{2-5j}$ ?

( 2021-12-06 20:26:18 +0200 )edit