If an affine algebra is defined over a field and has finite dimension considered as a vector space over its coefficient field, extra special operations are available on its elements.
Similar operations for affine algebras defined over general Euclidean rings will be supported in the future.
Given a finite dimensional affine algebra Q defined over a field, return the dimension of Q.
Given a finite dimensional affine algebra Q defined over a field, 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 defined over a field, 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 defined over a field, 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 defined over a field, return whether f is a unit.
Given an element f of a finite dimensional affine algebra Q defined over a field, 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 defined over a field, return the minimal polynomial of f as a univariate polynomial over the coefficient field of Q.
> Q := RationalField(); > A<x, y> := AffineAlgebra<Q, x, y | x^2 - 2, y^3 - 5>; > UP<z> := PolynomialRing(Q); > MinimalPolynomial(x + y); z^6 - 6*z^4 - 10*z^3 + 12*z^2 - 60*z + 17