# Affine Algebras

- [Introduction](introduction.md)

- [Creation of Affine Algebras](creation.md)

  - [`quo< P | J >: RngMPol, RngMPol → RngMPolRes`](creation.md#constructor-constructor-quo-rngmpol-rngmpol-rngmpolres)

  - [`quo< P | a₁, ..., aᵣ >: RngMPol, RngMPolElt, ..., RngMPolElt → RngMPolRes, Map`](creation.md#constructor-constructor-quo-rngmpol-rngmpolelt-rngmpolelt-rngmpolres-map)

  - [`P / J: RngMPol, RngMPol → RngMPolRes`](creation.md#operation-op-div-rngmpol-rngmpol)

  - [`AffineAlgebra< R, X | L >: Fld, List, List → RngMPolRes`](creation.md#constructor-constructor-affinealgebra-fld-list-list-rngmpolres)

  - [`Example: Creation`](creation.md#example-ex-3c15de)

- [Operations on Affine Algebras](operations.md)

  - [`Q . i: RngMPolRes, RngIntElt → RngMPolResElt`](operations.md#operation-operation-rngmpolres-rngintelt-rngmpolreselt)

  - [`CoefficientRing(Q): RngMPolRes → Rng`](operations.md#function-coefficientring-rngmpolres)

  - [`Rank(Q): RngMPolRes → RngIntElt`](operations.md#function-rank-rngmpolres)

  - [`DivisorIdeal(I): RngMPolRes → RngMPol`](operations.md#function-divisorideal-rngmpolres)

  - [`PreimageIdeal(I): RngMPolRes → RngMPol`](operations.md#function-preimageideal-rngmpolres)

  - [`PreimageRing(Q): RngMPolRes → RngMPol`](operations.md#function-preimagering-rngmpolres)

  - [`OriginalRing(Q): RngMPolRes → Rng`](operations.md#function-originalring-rngmpolres)

  - [`I eq J: RngMPolRes, RngMPolRes → BoolElt`](operations.md#operation-op-eq-rngmpolres-rngmpolres)

  - [`I subset J: RngMPolRes, RngMPolRes → BoolElt`](operations.md#operation-op-subset-rngmpolres-rngmpolres)

  - [`I + J: RngMPolRes, RngMPolRes → RngMPolRes`](operations.md#operation-op-plus-rngmpolres-rngmpolres)

  - [`I * J: RngMPolRes, RngMPolRes → RngMPolRes`](operations.md#operation-op-times-rngmpolres-rngmpolres)

  - [`I ^ n: RngMPolRes, RngIntElt → BoolElt`](operations.md#operation-op-pow-rngmpolres-rngintelt)

  - [`I meet J: RngMPolRes, RngMPolRes → RngMPolRes`](operations.md#operation-op-meet-rngmpolres-rngmpolres)

  - [`IsProper(I): RngMPolRes → BoolElt`](operations.md#function-isproper-rngmpolres)

  - [`IsZero(I): RngMPolRes → BoolElt`](operations.md#function-iszero-rngmpolres)

  - [`IsPrime(I): RngMPolRes → BoolElt`](operations.md#function-isprime-rngmpolres)

  - [`IsPrimary(I): RngMPolRes → BoolElt`](operations.md#function-isprimary-rngmpolres)

  - [`IsRadical(I): RngMPolRes → BoolElt`](operations.md#function-isradical-rngmpolres)

  - [`PrimaryDecomposition(I): RngMPolRes → [ RngMPolRes ], [ RngMPolRes ]`](operations.md#function-primarydecomposition-rngmpolres)

  - [`RadicalDecomposition(I): RngMPolRes → [ RngMPolRes ]`](operations.md#function-radicaldecomposition-rngmpolres)

  - [`Example: Operations`](operations.md#example-ex-f9694a)

- [Maps between Affine Algebras](affine-algebra-maps.md)

  - [`AffineAlgebraMapKernel(phi): Map → MPol`](affine-algebra-maps.md#function-affinealgebramapkernel-map)

- [Finite Dimensional Affine Algebras](finite-dimension-quotient.md)

  - [`HasFiniteDimension(Q): RngMPolRes → BoolElt`](finite-dimension-quotient.md#function-hasfinitedimension-rngmpolres)

  - [`Dimension(Q): RngMPolRes → RngIntElt`](finite-dimension-quotient.md#function-dimension-rngmpolres)

  - [`VectorSpace(Q): RngMPolRes → ModTupFld, Map`](finite-dimension-quotient.md#function-vectorspace-rngmpolres)

  - [`MonomialBasis(Q): RngMPolRes → [ RngMPolResElt ]`](finite-dimension-quotient.md#function-monomialbasis-rngmpolres)

  - [`MatrixAlgebra(Q): RngMPolRes → AlgMat, Map`](finite-dimension-quotient.md#function-matrixalgebra-rngmpolres)

  - [`RepresentationMatrix(f): RngMPolResElt → AlgMatElt`](finite-dimension-quotient.md#function-representationmatrix-rngmpolreselt)

  - [`IsUnit(f): RngMPolResElt → BoolElt`](finite-dimension-quotient.md#function-isunit-rngmpolreselt)

  - [`IsNilpotent(f): RngMPolResElt → BoolElt, RngIntElt`](finite-dimension-quotient.md#function-isnilpotent-rngmpolreselt)

  - [`MinimalPolynomial(f): RngMPolResElt → RngUPol`](finite-dimension-quotient.md#function-minimalpolynomial-rngmpolreselt)

  - [`Example: Minimal Polynomial`](finite-dimension-quotient.md#example-ex-caec0e)

- [Affine Algebras which are Fields](field.md)

  - [`Example: Elliptic Curve`](field.md#example-ex-b17854)

  - [`Example: Factorization`](field.md#example-ex-701393)

  - [`Example: Multi Extension`](field.md#example-ex-c88a64)

  - [`NumberField(A): RngMPolRes → FldNum, Map`](field.md#function-numberfield-rngmpolres)

  - [`Example: Number Field`](field.md#example-ex-7885e5)

- [Rings and Fields of Fractions of Affine Algebras](fractions.md)

  - [`RingOfFractions(Q): RngMPolRes → RngFunFrac`](fractions.md#function-ringoffractions-rngmpolres)

  - [`FieldOfFractions(Q): RngMPolRes → RngFunFrac`](fractions.md#function-fieldoffractions-rngmpolres)

  - [`Numerator(a): RngFunFracElt → RngMPolResElt`](fractions.md#function-numerator-rngfunfracelt)

  - [`Denominator(a): RngFunFracElt → RngMPolResElt`](fractions.md#function-denominator-rngfunfracelt)

  - [`Example: Field Of Fractions`](fractions.md#example-ex-a76faa)

  - [`Example: Extension`](fractions.md#example-ex-d7e7e8)