Finite Dimensional Affine Algebras

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.

HasFiniteDimension(Q): RngMPolRes -> BoolElt

Given an affine algebra \(Q\) defined over a field, return whether \(Q\) has finite dimension.

Dimension(Q): RngMPolRes -> RngIntElt

Given a finite dimensional affine algebra \(Q\) defined over a field, return the dimension of \(Q\).

VectorSpace(Q): RngMPolRes -> ModTupFld, Map

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 \(V\).

MonomialBasis(Q): RngMPolRes -> [ RngMPolResElt ]

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.

MatrixAlgebra(Q): RngMPolRes -> AlgMat, Map

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 \(A\).

RepresentationMatrix(f): RngMPolResElt -> AlgMatElt

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 \(f\).

IsUnit(f): RngMPolResElt -> BoolElt

Given an element \(f\) of a finite dimensional affine algebra \(Q\) defined over a field, return whether \(f\) is a unit.

IsNilpotent(f): RngMPolResElt -> BoolElt, RngIntElt

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 \(f^q = 0\).

MinimalPolynomial(f): RngMPolResElt -> RngUPol

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 \(Q\).

Example: Minimal Polynomial (ex-caec0e)

Suppose we wish to find the minimal polynomial of \(\theta = \sqrt 2 + \root 3 \of 5\) over \({\mathbb{Q}}\). To do this we can just compute the minimal polynomial of (the coset of) \(x + y\) over \({\mathbb{Q}}\) 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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