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# 10 Nearfields, planar nearrings and weakly divisible nearrings

### Sections

1. Dickson numbers
2. Dickson nearfields
3. Exceptional nearfields
4. Planar nearrings
5. Weakly divisible nearrings

A nearfield is a nearring with 1 where each nonzero element has a multiplicative inverse. The (additive) group reduct of a finite nearfield is necessarily elementary abelian. For an exposition of nearfields we refer to <[>Waehling:Fastkoerper].

Let (N,+,·) be a left nearring. For a,b ∈ N we define a ≡ b iff a·n = b·n for all n ∈ N. If a ≡ b, then a and b are called equivalent multipliers. A nearring N is called planar if | N/ ≡ | ≥ 3 and if for any two non-equivalent multipliers a and b in N, for any c ∈ N, the equation a·x = b·x + c has a unique solution. See <[>Clay:Nearrings] for basic results on planar nearrings.

All finite nearfields are planar nearrings.

A left nearring (N,+,·) is called weakly divisible if ∀a,b ∈ N ∃x ∈ N : a·x = b or b·x = a.

All finite integral planar nearrings are weakly divisible.

## 10.1 Dickson numbers

- `IsPairOfDicksonNumbers(` q`,` n`)`

  A pair of Dickson numbers (q,n) consists of a prime power integer q and a natural number n such that for p = 4 or p prime, p|n implies p|q−1.

          gap> IsPairOfDicksonNumbers( 5, 4 );
          true

  ## 10.2 Dickson nearfields

- `DicksonNearFields(` q`,` n`)`

  All finite nearfields with 7 exceptions can be obtained via socalled coupling maps from finite fields. These nearfields are called Dickson nearfields.

  The multiplication map of such a Dickson nearfield is given by a pair of Dickson numbers (q,n) in the following way:

  Let F = GF(qn) and w be a primitive element of F. Let H be the subgroup of (F\{0},·) generated by wn. Then {w(qi−1)/(q−1) | 0 ≤ i ≤ n−1 } is a set of coset representatives of H in F\{0}. For f ∈ Hw(qi−1)/(q−1) and x ∈ F define f*x = f·xqi and 0*x = 0. Then * is a nearfield multiplication on the additive group (F,+).

  Note that a Dickson nearfield is not uniquely determined by (q,n), since w can be chosen arbitrarily. Different choices of w may yield isomorphic nearfields.

  `DicksonNearFields` returns a list of the non-isomorphic Dickson nearfields determined by the pair of Dickson numbers (q,n)

          gap> DicksonNearFields( 5, 4 );
          [ ExplicitMultiplicationNearRing ( <pc group of size 625 with
              4 generators> , multiplication ),
            ExplicitMultiplicationNearRing ( <pc group of size 625 with
              4 generators> , multiplication ) ]

- `NumberOfDicksonNearFields(` q`,` n`)`

  `NumberOfDicksonNearFields` returns the number of non-isomorphic Dickson nearfields which can be obtained from a pair of Dickson numbers (q,n). This number is given by Φ(n)/k. Here Φ(n) denotes the number of relatively prime residues modulo n and k is the multiplicative order of p modulo n where p is the prime divisor of q.

          gap> NumberOfDicksonNearFields( 5, 4 );
          2

  ## 10.3 Exceptional nearfields

- `ExceptionalNearFields(` q`)`

  There are 7 finite nearfields which cannot be obtained from finite fields via a Dickson process. They are of size p2 for p = 5, 7, 11, 11, 23, 29, 59. (There exist 2 exceptional nearfields of size 121.)

  `ExceptionalNearFields` returns the list of exceptional nearfields for a given size q.

          gap> ExceptionalNearFields( 25 );
          [ ExplicitMultiplicationNearRing ( <pc group of size 25 with
              2 generators> , multiplication ) ]

- `AllExceptionalNearFields()`

  There are 7 finite nearfields which cannot be obtained from finite fields via a Dickson process. They are of size p2 for p = 5, 7, 11, 11, 23, 29, 59. (There exist 2 exceptional nearfields of size 121.)

  `AllExceptionalNearFields` without argument returns the list of exceptional nearfields.

          gap> AllExceptionalNearFields();
          [ ExplicitMultiplicationNearRing ( <pc group of size 25 with
              2 generators> , multiplication ),
            ExplicitMultiplicationNearRing ( <pc group of size 49 with
              2 generators> , multiplication ),
            ExplicitMultiplicationNearRing ( <pc group of size 121 with
              2 generators> , multiplication ),
            ExplicitMultiplicationNearRing ( <pc group of size 121 with
              2 generators> , multiplication ),
            ExplicitMultiplicationNearRing ( <pc group of size 529 with
              2 generators> , multiplication ),
            ExplicitMultiplicationNearRing ( <pc group of size 841 with
              2 generators> , multiplication ),
            ExplicitMultiplicationNearRing ( <pc group of size 3481 with
              2 generators> , multiplication ) ]

  ## 10.4 Planar nearrings

- `PlanarNearRing(` G`,` phi`,` reps`)`

  A finite Ferrero pair is a pair of groups (N,Φ) where Φ is a fixed-point-free automorphism group of (N,+).

  Starting with a Ferrero pair (N,Φ) we can construct a planar nearring in the following way, <[>Clay:Nearrings:] Select representatives, say e1,…,et, for some or all of the non-trivial orbits of N under Φ. Let C = Φ(e1)∪…∪Φ(et). For each x ∈ N we define a * x = 0 for a ∈ N\C, and a * x=ϕa(x) for a ∈ Φ(ei) ⊂ C and ϕa(ei)=a. Then (N,+,*) is a (left) planar nearring.

  Every finite planar nearring can be constructed from some Ferrero pair together with a set of orbit representatives in this way.

  `PlanarNearRing` returns the planar nearring on the group G determined by the fixed-point-free automorphism group phi and the list of chosen orbit representatives reps.

          gap> C7 := CyclicGroup( 7 );;
          gap> i := GroupHomomorphismByFunction( C7, C7, x -> x^-1 );;
          gap> phi := Group( i );;
          gap> orbs := Orbits( phi, C7 );
          [ [ <identity> of ... ], [ f1, f1^6 ], [ f1^2, f1^5 ],
            [ f1^3, f1^4 ] ]
          gap> # choose reps from the orbits
          gap> reps := [orbs[2][1], orbs[3][2]];
          [ f1, f1^5 ]
          gap> n := PlanarNearRing( C7, phi, reps );
          ExplicitMultiplicationNearRing ( <pc group of size 7 with
          1 generator> , multiplication )

- `OrbitRepresentativesForPlanarNearRing(` G`,` phi`,` i`)`

  Let (N,Φ) be a Ferrero pair, and let E = { e1,…,es } and F = { f1,…,ft } be two sets of non-zero orbit representatives. The nearring obtained from N,Φ, E by the Ferrero construction (see `PlanarNearRing`) is isomorphic to the nearring obtained from N,Φ, F iff there exists an automorphism α of (N,+) that normalizes Φ such that { α(e1),…,α(es) } = { f1,…,ft }.

  The function `OrbitRepresentativesForPlanarNearRing` returns precisely one set of representatives of cardinality i for each isomorphism class of planar nearrings which can be generated from the Ferrero pair ( G, phi ).

          gap> C7 := CyclicGroup( 7 );;
          gap> i := GroupHomomorphismByFunction( C7, C7, x -> x^-1 );;
          gap> phi := Group( i );;
          gap> reps := OrbitRepresentativesForPlanarNearRing( C7, phi, 2 );
          [ [ f1, f1^2 ], [ f1, f1^5 ] ]
          gap> n1 := PlanarNearRing( C7, phi, reps[1] );;
          gap> n2 := PlanarNearRing( C7, phi, reps[2] );;
          gap> IsIsomorphicNearRing( n1, n2 );
          false

  ## 10.5 Weakly divisible nearrings

- `WdNearRing(` G`,` psi`,` phi`,` reps`)`

  Every finite (left) weakly divisible nearring (N,+,·) can be constructed in the following way:

  (1) Let ψ be an endomorphism of the group (N,+) such that Ker ψ = Image ψr−1 for some integer r, r > 0. (Let ψ0 := id.)

  (2) Let Φ be an automorphism group of (N,+) such that ψΦ ⊆ Φψ and Φ acts fixed-point-free on N\ Image ψ. (That is, for each φ ∈ Φ there exists φ′ ∈ Φ such that ψφ = φ′ψ and for all n ∈ N\ Image ψ the equality nφ = n implies φ = id. Note that our functions operate from the right just like GAP-mappings do.)

  (3) Let E ⊆ N be a complete set of orbit representatives for Φ on N\ Image ψ, such that for all e1, e2 ∈ E, for all φ ∈ Φ and for all 1 ≤ i ≤ r−1 the equality e1φψi = e2ψi implies φψi = ψi.

  Then for all n ∈ N, n ≠ 0, there are i ≥ 0 ,φ ∈ Φ and e ∈ E such that n = eφψi; furthermore, for fixed n, the endomorphism φψi is independent of the choice of e and φ in the representation of n.

  For all x ∈ N, e ∈ E,φ ∈ Φ and i ≥ 0 define 0·x : = 0 and

  eφψi·x : = xφψi

  Then (N,+,·) is a zerosymmetric (left) wd nearring.

  `WdNearRing` returns the wd nearring on the group G as defined above by the nilpotent endomorphism psi, the automorphism group phi and a list of orbit representatives reps where the arguments fulfill the conditions (1) to (3).

          gap> C9 := CyclicGroup( 9 );;
          gap> psi := GroupHomomorphismByFunction( C9, C9, x -> x^3 );;
          gap> Image( psi );
          Group([ f2, <identity> of ... ])
          gap> Image( psi ) = Kernel( psi );
          true
          gap> a := GroupHomomorphismByFunction( C9, C9, x -> x^4 );;
          gap> phi := Group( a );;
          gap> Size( phi );
          3
          gap> orbs := Orbits( phi, C9 );
          [ [ <identity> of ... ], [ f2 ], [ f2^2 ], [ f1, f1*f2, f1*f2^2 ],
            [ f1^2, f1^2*f2^2, f1^2*f2 ] ]
          gap> # choose reps from the orbits outside of Image( psi )
          gap> reps := [orbs[4][1], orbs[5][1]];
          [ f1, f1^2 ]
          gap> n := WdNearRing( C9, psi, phi, reps );
          ExplicitMultiplicationNearRing ( <pc group of size 9 with
          2 generators> , multiplication )

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