ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 30 Nov 2018 02:06:30 +0100Reduction In Quotient Rings (x^2 - x = 0)https://ask.sagemath.org/question/44514/reduction-in-quotient-rings-x2-x-0/Hi,
Thank you for taking the time to read this, it is very much appreciated.
My question concerns how to ensure that a polynomial within a quotient ring has the following property:
(x^2)k = 0
whereby x is any variable in the quotient ring and k is a positive integer.
This is the way I tried to go about the situation: I created a polynomial ring
P.<x,y,z,w> = PolynomialRing(GF(2), 4, order = 'degrevlex')
Since I am not working within a quotient ring, x^2 (or any of the other three variables) does not 'become' 0. Since I would like the property of x^2 = 0, I decided to create a quotient ring with some field equations:
Q = P.quotient_ring(ideal([var**q - var for var in P.gens()]))
whereby `q = P.base_ring.order()` .
However, when I then created the following polynomial, its parent was still P, so I changed its ring:
f1 = y*z + y*w + w^2
f1 = f1.change_ring(Q)
However, when I print f1, it, w^2 is still w^2 and has not reduced down to 0. I was wondering if I am missing something, please? This gets annoying because I am going to be working with Macaulay Matrices and hence, it is essential that I work within a quotient ring. Maybe I am missing some mathematics since this is all very new to me...
This is my sage input:
sage: P.<x,y,z,w> = PolynomialRing(GF(2), 4, order = 'degrevlex')
sage: q = P.base_ring().order()
sage: Q = P.quotient_ring(ideal([var**q - var for var in P.gens()]))
sage: f1 = y*z + y*w + w^2
sage: f1
y*z + y*w + w^2
sage: f1 = f1.change_ring(Q)
sage: f1
y*z + y*w + w^2
How would go about to ensure that w^2 = 0? I've already tried adding the original polynomial to the field equations when creating the quotient ring and changing its ring afterwards, like so:
sage: P.<x,y,z,w> = PolynomialRing(GF(2), 4, order = 'degrevlex')
sage: q = P.base_ring().order()
sage: f1 = y*z + y*w + w^2
sage: Q = P.quotient_ring(ideal([f1] + [var**q - var for var in P.gens()]))
sage: f1 = f1.change_ring(Q)
sage: f1
y*z + y*w + w^2
But as you can see, nothing happened...
Thank you!JoaoDDuarteFri, 30 Nov 2018 02:06:30 +0100https://ask.sagemath.org/question/44514/