# Revision history [back]

It is doable as follows. Assuming that y^2 = x^3 + x +1 is like working in a quotient polynomial ring modulo the ideal generated by -y^2 + x^3 + x +1. To be able to make polynomial divisions, you just have to extend your quotient polynomial ring into its fraction field:

sage: R = PolynomialRing(QQ,'x,y') ; R
Multivariate Polynomial Ring in x, y over Rational Field
sage: R.inject_variables()
Defining x, y
sage: I = R.ideal([-y^2 + x^3 + x +1]) ; I
Ideal (x^3 - y^2 + x + 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: Q = R.quotient(I) ; Q
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^3 - y^2 + x + 1)
sage: F = Q.fraction_field() ; F
Fraction Field of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^3 - y^2 + x + 1)
sage: F((x^3 + 2*x + 1)/y)
(ybar^2 + xbar)/ybar