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I am working in the following field extension:

p = 2**64-2**32+1
P.<X,A,B,C> = PolynomialRing(GF(p))
GF.<x,a,b,c> = P.quotient_ring(X^3-X-1)

My objective is to take a generic element of this field p(x) = a + b*x + c*x^2 and compute its inverse, where its coefficients are a function of a,b,c.

However, if I try to directly do it:

q = x^2
qinv = 1 / q

I get a singular error. I get an error (ArithmeticError: Division failed. The numerator is not a multiple of the denominator.) if I change the prime to, say, 5. If I do not use the variables to express the generic coefficients of p(x), then clearly it works, but I do not get the desired result.

Is there any way I can achieve what I need?

I am working in the following field extension:

p = 2**64-2**32+1
P.<X,A,B,C> = PolynomialRing(GF(p))
GF.<x,a,b,c> = P.quotient_ring(X^3-X-1)

My objective is to take a generic element of this field p(x) = a + b*x + c*x^2 and compute its inverse, where its coefficients are a function of a,b,c.

However, if I try to directly do it:

q = x^2
qinv = 1 / q

I get a singular error. I get an error (ArithmeticError: Division failed. The numerator is not a multiple of the denominator.) if I change the prime to, say, 5. If I do not use the variables to express the generic coefficients of p(x), then clearly it works, but I do not get the desired result.

Is there any way I can achieve what I need?