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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 an affine algebra Q defined over a field,
return whether Q has finite dimension.
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,
return the basis B of monomials of Q. This is a sequence
of monomials in Q of length d, such that the image f(B[i])=V.i
where V and f are the return values of VectorSpace above.
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 fq = 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.
Suppose we wish to find the minimal polynomial of
θ = √2 + root 3 of 5 over Q. To do this we can
just compute the minimal polynomial of (the coset of)
x + y over Q in the affine algebra Q[x, y]/(x 2 - 2, y 3 - 5).
> 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
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