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Polynomial ring over the ring of integers modulo 3

asked 2020-05-08 07:37:31 -0600

PedroFreitas90 gravatar image

updated 2020-05-08 16:06:27 -0600

slelievre gravatar image

I am using SageMath and I want to use a polynomial ring in this form: (Z/3)[x]/(x^p - x - 1) , with Z as polynomial ring over integer Ring. How can I build that? Until now I only could declare Z[x]/x^p - x - 1

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answered 2020-05-08 09:39:34 -0600

rburing gravatar image

The most literal interpretation is to build the quotient ring $(\mathbb{Z}/3\mathbb{Z})[x]/(x^p - x - 1)$:

sage: p = 3
sage: A.<x> = PolynomialRing(Zmod(3))
sage: B.<y> = A.quotient(x^p - x - 1)
sage: B.cardinality().factor()
3^3
sage: B.is_field()
True

Since $3$ is prime you can also replace Zmod(3) by GF(3).

If $x^p - x - 1$ is irreducible (for example for $p=3$) then it is a modulus for the field with $3^p$ elements:

sage: C.<z> = GF(3^p, modulus=x^p - x - 1)
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Asked: 2020-05-08 07:37:31 -0600

Seen: 58 times

Last updated: May 08