# Revision history [back]

Over QQ you can do that

sage: R = PolynomialRing(QQ, 'x')
sage: x = R.gen()
sage: quot = R.quotient_by_principal_ideal(x**7-1)
sage: y = quot.gen()
sage: 1/(y**6-y**4+y**3-y**2+1)
2/13*xbar^6 - 1/13*xbar^5 + 3/13*xbar^4 - 1/13*xbar^3 + 2/13*xbar^2 + 4/13*xbar + 4/13


and you can see that the inverse does not exist over ZZ, as 13 must be invertible

Over QQ you can do that

sage: R = PolynomialRing(QQ, 'x')
sage: x = R.gen()
sage: quot = R.quotient_by_principal_ideal(x**7-1)
sage: y = quot.gen()
sage: 1/(y**6-y**4+y**3-y**2+1)
2/13*xbar^6 - 1/13*xbar^5 + 3/13*xbar^4 - 1/13*xbar^3 + 2/13*xbar^2 + 4/13*xbar + 4/13
sage: 1/(y**6-y**4+y**3+y**2-1)
16/43*xbar^6 + 11/43*xbar^5 + 21/43*xbar^4 + 1/43*xbar^3 - 2/43*xbar^2 + 4/43*xbar - 8/43


and you can see that the inverse does inverses do not exist over ZZ, as 13 or 43 must be invertible