### Polynomial division on non usual ring.

Hello everyone, I hope you are well.

I'm trying to do the algorithm of division between polynomials in an unusual ring and I'm having problems. I searched in some forums and couldn't find anything on the subject.

The idea is to do the division in the following ring: R.<x,y> = ~~PolynomialRing(GF(),2,'x','y', order='lex').~~PolynomialRing(UCF(),2,'x','y', order='lex'), where UCF = UniversalCyclotomicField().

The idea is to get the coefficient between f/g=q, with f, g \in R.<x,y> .

But overall the result of f/g is just giving f/g, even in the case where f=g.

I try this method in the link: https:// ask . sagemath . org/question/50406/polynomial-division-in-quotient-rings /

but sage can't do the math

Thank you for your attention.