NEARFIELDS

DETAILS

 
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 FldNear_dicksonpairs (H22E1)
            NumberOfVariants(q, v) : RngIntElt, RngIntElt -> RngIntElt
            NumberOfVariants(N) : NfdDck -> RngIntElt
            VariantRepresentatives(q, v) : RngIntElt, RngIntElt -> SeqEnum
            Example FldNear_variants (H22E2)
            DicksonNearfield(q, v : parameters) : RngIntElt, RngIntElt -> NfdDck
            Example FldNear_dickson (H22E3)

      Zassenhaus Nearfields
            ZassenhausNearfield(n) : RngIntElt -> NfdZss
            Example FldNear_zassenhaus (H22E4)

 
Operations on Elements

      Nearfield Arithmetic
            Inverse(a) : NfdElt -> NfdElt

      Equality and Membership

      Parent and Category
            N ! x : Nfd, FldFinElt -> NfdElt
            ElementToSequence(x) : NfdElt) -> SeqEnum

      Predicates on Nearfield Elements
            Example FldNear_simplearith (H22E5)
            Example FldNear_leftdist (H22E6)

 
Operations on Nearfields
      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
      Example FldNear_unitgrp (H22E7)
      Order(x) : NfdElt -> RngIntElt
      AffineGroup(N) : Nfd -> GrpMat
      ExtendedUnitGroup(D) : NfdDck -> GrpMat

 
Automorphisms
      IsIsomorphic(N1, N2) : NfdDck, NfdDck -> BoolElt, Map
      AutomorphismGroup(N) : NfdDck -> GrpPerm, Map

 
Nearfield Planes
      ProjectivePlane( N : parameters) : Nfd -> PlaneProj, PlanePtSet, PlaneLnSet
      Example FldNear_projplane (H22E8)

      Hughes Planes
            HughesPlane(N : parameters) : Nfd -> PlaneProj, PlanePtSet, PlaneLnSet
            Example FldNear_hughes (H22E9)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013