Nearfields

• Introduction
• Nearfield Properties
  • Sharply Doubly Transitive Groups
• Constructing Nearfields
  • Dickson Nearfields
    • DicksonPairs(p, hlo, hhi, vlo, vhi): RngIntElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → SeqEnum
    • DicksonPairs(p, h1, v1): RngIntElt, RngIntElt, RngIntElt → SeqEnum
    • DicksonTriples(p, hb, vb): RngIntElt, RngIntElt, RngIntElt → SeqEnum
    • Example: Dickson Pairs
    • NumberOfVariants(q, v): RngIntElt, RngIntElt → RngIntElt
    • NumberOfVariants(N): NfdDck → RngIntElt
    • VariantRepresentatives(q, v): RngIntElt, RngIntElt → SeqEnum
    • Example: Variants of a Dickson Pair
    • DicksonNearfield(q, v : parameters): RngIntElt, RngIntElt → NfdDck
    • Example: Constructing Dickson Nearfields
  • Zassenhaus Nearfields
    • ZassenhausNearfield(n): RngIntElt → NfdZss
    • Example: Zassenhaus Nearfields
• Operations on Elements
  • Nearfield Arithmetic
    • + a: NfdElt → NfdElt
    • - a: NfdElt → NfdElt
    • a + b: NfdElt, NfdElt → NfdElt
    • a - b: NfdElt, NfdElt → NfdElt
    • a * b: NfdElt, NfdElt → NfdElt
    • a / b: NfdElt, NfdElt → NfdElt
    • a ^ k: NfdElt, RngIntElt → NfdElt
    • a +:= b: NfdElt, NfdElt → NfdElt
    • a -:= b: NfdElt, NfdElt → NfdElt
    • a *:= b: NfdElt, NfdElt → NfdElt
    • Inverse(a): NfdElt → NfdElt
  • Equality and Membership
    • a eq b: NfdElt, NfdElt → BoolElt
    • a ne b: NfdElt, NfdElt → BoolElt
    • a in N: NfdElt, Rng → BoolElt
    • a notin N: NfdElt, Rng → BoolElt
  • Parent and Category
    • Parent(a): NfdElt → FldFin
    • Category(a): NfdElt → Cat
    • N ! x: Nfd, FldFinElt → NfdElt
    • Element(N, x): Nfd, FldFinElt → NfdElt
    • ElementToSequence(x): NfdElt → SeqEnum
  • Predicates on Nearfield Elements
    • IsZero(a): NfdElt → BoolElt
    • IsUnit(a): NfdElt → BoolElt
    • IsIdentity(a): NfdElt → BoolElt
    • Example: Simple Nearfield Arithmetic
    • Example: Failure of Left Distributivity
• Operations on Nearfields
  • N eq M: NfdDck, NfdDck → BoolElt
  • N ne M: NfdDck, NfdDck → BoolElt
  • #N: Nfd → RngIntElt
  • Cardinality(N): Nfd → RngIntElt
  • Random(N): Nfd → NfdElt
  • Identity(N): Nfd → NfdElt
  • Zero(N): Nfd → NfdElt
  • PrimeField(N): Nfd → FldFin
  • Kernel(N): Nfd → FldFin
• The Group of Units
  • UnitGroup(N): Nfd → GrpMat, Map
  • UnitGroup(GrpPerm, N): Nfd → GrpPerm
  • UnitGroup(GrpPC, N): NfdDck → GrpPC
  • UnitGroup(GrpPC, N): NfdZss → GrpPC
  • Example: Unit Group of a Subnearfield
  • Order(x): NfdElt → RngIntElt
  • AffineGroup(N): Nfd → GrpMat
  • AffineGroup(GrpPerm, N): Nfd → GrpPerm
  • AffineGroup(GrpPC, N): NfdDck → GrpPC
  • AffineGroup(GrpPC, N): NfdZss → GrpPC
  • ExtendedUnitGroup(D): NfdDck → GrpMat
• Automorphisms
  • IsIsomorphic(N1, N2): NfdDck, NfdDck → BoolElt, Map
  • AutomorphismGroup(N): NfdDck → GrpPerm, Map
  • AutomorphismGroup(N): NfdZss → GrpPerm
• Nearfield Planes
  • ProjectivePlane(N : parameters): Nfd → PlaneProj, PlanePtSet, PlaneLnSet
  • Example: Projective Plane of a Nearfield
  • Hughes Planes
    • HughesPlane(N : parameters): Nfd → PlaneProj, PlanePtSet, PlaneLnSet
    • Example: Hughes Planes
• Bibliography