Affine Algebras

• Introduction
• Creation of Affine Algebras
  • quo< P | J >: RngMPol, RngMPol → RngMPolRes
  • quo< P | a₁, ..., aᵣ >: RngMPol, RngMPolElt, ..., RngMPolElt → RngMPolRes, Map
  • P / J: RngMPol, RngMPol → RngMPolRes
  • AffineAlgebra< R, X | L >: Fld, List, List → RngMPolRes
  • Example: Number Field Extensions as Affine Algebras
• Operations on Affine Algebras
  • Q . i: RngMPolRes, RngIntElt → RngMPolResElt
  • CoefficientRing(Q): RngMPolRes → Rng
  • Rank(Q): RngMPolRes → RngIntElt
  • DivisorIdeal(I): RngMPolRes → RngMPol
  • PreimageIdeal(I): RngMPolRes → RngMPol
  • PreimageRing(Q): RngMPolRes → RngMPol
  • OriginalRing(Q): RngMPolRes → Rng
  • I eq J: RngMPolRes, RngMPolRes → BoolElt
  • I subset J: RngMPolRes, RngMPolRes → BoolElt
  • I + J: RngMPolRes, RngMPolRes → RngMPolRes
  • I * J: RngMPolRes, RngMPolRes → RngMPolRes
  • I ^ n: RngMPolRes, RngIntElt → BoolElt
  • I meet J: RngMPolRes, RngMPolRes → RngMPolRes
  • IsProper(I): RngMPolRes → BoolElt
  • IsZero(I): RngMPolRes → BoolElt
  • IsPrime(I): RngMPolRes → BoolElt
  • IsPrimary(I): RngMPolRes → BoolElt
  • IsRadical(I): RngMPolRes → BoolElt
  • PrimaryDecomposition(I): RngMPolRes → [ RngMPolRes ], [ RngMPolRes ]
  • RadicalDecomposition(I): RngMPolRes → [ RngMPolRes ]
  • Example: Ideals of Affine Algebras
• Maps between Affine Algebras
  • AffineAlgebraMapKernel(phi): Map → MPol
• Finite Dimensional Affine Algebras
  • HasFiniteDimension(Q): RngMPolRes → BoolElt
  • Dimension(Q): RngMPolRes → RngIntElt
  • VectorSpace(Q): RngMPolRes → ModTupFld, Map
  • MonomialBasis(Q): RngMPolRes → [ RngMPolResElt ]
  • MatrixAlgebra(Q): RngMPolRes → AlgMat, Map
  • RepresentationMatrix(f): RngMPolResElt → AlgMatElt
  • IsUnit(f): RngMPolResElt → BoolElt
  • IsNilpotent(f): RngMPolResElt → BoolElt, RngIntElt
  • MinimalPolynomial(f): RngMPolResElt → RngUPol
  • Example: Minimal Polynomial of an Algebraic Number
• Affine Algebras which are Fields
  • Example: Elliptic Curve Function Field
  • Example: Factorization over an Affine Algebra
  • Example: Multiple Field Extension
  • NumberField(A): RngMPolRes → FldNum, Map
  • Example: Number Field of Degree 8
• Rings and Fields of Fractions of Affine Algebras
  • RingOfFractions(Q): RngMPolRes → RngFunFrac
  • FieldOfFractions(Q): RngMPolRes → RngFunFrac
  • Numerator(a): RngFunFracElt → RngMPolResElt
  • Denominator(a): RngFunFracElt → RngMPolResElt
  • Example: Field of Fractions
  • Example: Extension and Contraction of Ideals