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Units in a quotient of a polynomial ring

asked 2014-08-30 09:47:58 -0500

Chebolu gravatar image

updated 2014-08-30 09:49:17 -0500

How can I do calculations in a quotient ring using sage? For example, I would like to find all the units (or at least count the number of units) in this quotient ring. Z_4[x]/(x^4+1, 2+2x). (Here Z_4 is the ring of integers mod 4.) Is that possible? I could not find any useful documentation for quotients of polynomial rings that are not a PID. Any help would be greatly appreciated. Thank you.

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answered 2014-08-30 10:13:31 -0500

tmonteil gravatar image

updated 2014-08-30 12:26:22 -0500

I think the following is self-explanatory:

sage: Z_4 = IntegerModRing(4)
sage: Z_4
Ring of integers modulo 4
sage: P.<x> = PolynomialRing(Z_4)
sage: P
Univariate Polynomial Ring in x over Ring of integers modulo 4
sage: I = P.ideal(x^4+1, 2+2*x)
sage: I
Ideal (x^4 + 1, 2*x + 2) of Univariate Polynomial Ring in x over Ring of integers modulo 4
sage: R = P.quotient(I)
sage: R
Quotient of Univariate Polynomial Ring in x over Ring of integers modulo 4 by the ideal (x^4 + 1, 2*x + 2)

The problem afterwards is that Sage is does not know much about R, for example, even if R is finite, you can not iterate over it:

sage: R.list()
NotImplementedError: object does not support iteration

It does not even knows that R is finite:

sage: R.is_finite()
NotImplementedError:

Nor whether some elements are units:

sage: a = R.an_element()
sage: a
xbar
sage: a.is_unit()
NotImplementedError:

But you are welcome to join and help fill to the gap !

EDIT What makes me fear however is the following:

sage: 2*xbar+2 == 0
False
sage: (2*xbar+2 ).is_zero()
NotImplementedError:

We should definitely prevent Sage answering random things when it does not know.

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Comments

Thank you for your help. That is exactly the problem I have been facing. I am able to input the ring but unable to make sage do any useful calculations with it. So in a way it is useless. I am trying to find a way around this problem but I don't see how to this yet.

Chebolu gravatar imageChebolu ( 2014-08-30 10:32:36 -0500 )edit

Ok very good. Once you know about this, please do not hesitate to share your investigations so that we will be able to include those in Sage.

tmonteil gravatar imagetmonteil ( 2014-08-30 10:56:11 -0500 )edit

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Asked: 2014-08-30 09:47:58 -0500

Seen: 251 times

Last updated: Aug 30 '14