Magma allows one to create the quotient ring of a multivariate polynomial ring P over a field by an ideal J of P. Such quotient rings are known as affine algebras. If an affine algebra has finite dimension considered as a vector space over the coefficient field, extra special operations are available on its elements.
Given a multivariate polynomial ring P (over a field) and an ideal J of P, return the quotient ring P/J. The ideal J may either be specified as an ideal or by a list a_1, a_2, ..., a_r, of generators which all lie in P. 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, I.3^2 + I.4 >;.
Given a multivariate polynomial ring P (over a field) and an ideal J of P, return the quotient affine algebra P/J.
Given an affine algebra Q together with a list L = a_1, a_2, ..., a_r of elements of Q, return the ideal of Q generated by L.
This subsection describes operations on affine rings. Most of the operations
are very similar to those for multivariate polynomial rings; such operations
are done by mapping the computation to the preimage ideal and then by
mapping the result back into the affine algebra. See the corresponding
functions for the multivariate polynomial rings for details.
Q . i : RngMPolRes, RngIntElt -> RngMPolResElt
Given an affine algebra Q, return the i-th indeterminate of Q as an element of Q.
Return the coefficient ring of the affine algebra Q.
Return the rank of the affine algebra Q (the number of indeterminates of Q).
Given an ideal I of an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the ideal J.
Given an ideal I of an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the ideal I' of P such that the image of I' under the natural epimorphism P -> Q is I.
Given an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the polynomial ring P.
Given two ideals I and J of the same affine algebra Q, return true if and only if I and J are equal.
Given two ideals I and J of the same affine algebra Q, return true if and only if I is contained in J.
Given two ideals I and J of the same affine algebra Q, return the sum I + J.
Given two ideals I and J of the same affine algebra Q, return the product I * J.
Given an ideal I of an affine algebra Q and an integer n, return the power I^n.
Given two ideals I and J of the same affine algebra Q, return the intersection Iintersect J.
Given two ideals I and J of the same affine algebra Q, return the colon ideal I:J.
Given an ideal I of the affine algebra Q, return whether I is proper; that is, whether I is strictly contained in Q.
Given an ideal I of the affine algebra Q, return whether I is the zero ideal. Note that this is equivalent to whether the preimage ideal of I is the divisor ideal of Q.
Given an ideal I of the affine algebra Q, return whether I is a prime ideal.
Given an ideal I of the affine algebra Q, return whether I is a primary ideal.
Given an ideal I of the affine algebra Q, return whether I is a radical ideal.
Given an ideal I of the affine algebra Q, return the primary decomposition of I, together with the associated primes.
Given an ideal I of the affine algebra Q, return the (prime) decomposition of the radical of I.
If an affine algebra has finite dimension considered as a vector
space over its coefficient field, extra special operations are
available on its elements.
Dimension(Q) : RngMPolRes -> RngIntElt
Given a finite dimensional affine algebra Q, return the dimension of Q.
Given a finite dimensional affine algebra Q, construct the vector space V isomorphic to Q, and return V together with the isomorphism f from Q onto V.
Given a finite dimensional affine algebra Q, construct the matrix algebra A isomorphic to Q, and return A together with the isomorphism f from Q onto A.
Given an element f of a finite dimensional affine algebra 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 affine algebra Q, return whether f is a unit.
Given an element f of a finite dimensional affine algebra 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 affine algebra 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()); > MinimalPolynomial(a + b); z^6 - 6*z^4 - 10*z^3 + 12*z^2 - 60*z + 17