Currently it is only possible to create quotient rings in multivariate polynomial rings over fields. If the quotient ring has finite dimension (considered as a vector space over the coefficient field), further operations are available on its elements.
Given an ideal I in the multivariate polynomial ring R (over a field), return the quotient R/I. The ideal I may either be specified as an ideal or by a list a_1, a_2, ..., a_r, of generators. The angle bracket notation can be used to assign names to the indeterminates: Q<q, r> := quo< I | I.1 + I.2, I.2^2 - 2 >;.
Given a quotient Q of a multivariate polynomial ring R, return the i-th indeterminate of Q as an element of Q.
Return the coefficient ring of polynomial quotient ring Q.
Return the rank of polynomial quotient ring Q (the number of indeterminates of Q).
If a quotient ring has finite dimension (considered as a vector space over its
coefficient field), further operations are available on its elements.
Dimension(Q) : RngQPol -> RngIntElt
Given a finite dimensional quotient ring Q, return the dimension of Q.
Given an element f of a finite dimensional quotient ring Q, return the representation matrix of f, which is a d by d matrix over the coefficient field of Q (where d is the dimension of Q) which represents f.
Given an element f of a finite dimensional quotient ring Q, return whether f is a unit.
Given an element f of a finite dimensional quotient ring Q, return whether f is nilpotent, and if so, return also the smallest q such that f^q = 0.
Given an element f of a finite dimensional quotient ring Q, return the minimal polynomial of f as a univariate polynomial over the coefficient field of Q.
> P<x, y> := PolynomialRing(RationalField(), 2); > Q<a, b> := quo<P | x^2 - 2, y^3 - 5>; > UP<z> := PolynomialRing(RationalField()); > print MinimalPolynomial(a + b); z^6 - 6*z^4 - 10*z^3 + 12*z^2 - 60*z + 17