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<h1>10 Nearfields, planar nearrings and weakly divisible nearrings</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP010.htm#SECT001">Dickson numbers</a>
<li> <A HREF="CHAP010.htm#SECT002">Dickson nearfields</a>
<li> <A HREF="CHAP010.htm#SECT003">Exceptional nearfields</a>
<li> <A HREF="CHAP010.htm#SECT004">Planar nearrings</a>
<li> <A HREF="CHAP010.htm#SECT005">Weakly divisible nearrings</a>
</ol>

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 <a href="biblio.htm#Waehling:Fastkoerper"><[>Waehling:Fastkoerper</cite></a>].

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 <a href="biblio.htm#Clay:Nearrings"><[>Clay:Nearrings</cite></a>] 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.


<h2><a name="SECT001">10.1 Dickson numbers</a></h2>

<a name = "SSEC001.1"></a>
<li><code>IsPairOfDicksonNumbers( </code><var>q</var><code>, </code><var>n</var><code> )</code>

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.

<pre>
 gap> IsPairOfDicksonNumbers( 5, 4 );
 true
</pre>


<h2><a name="SECT002">10.2 Dickson nearfields</a></h2>

<a name = "SSEC002.1"></a>
<li><code>DicksonNearFields( </code><var>q</var><code>, </code><var>n</var><code> )</code>

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.

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

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

<a name = "SSEC002.2"></a>
<li><code>NumberOfDicksonNearFields( </code><var>q</var><code>, </code><var>n</var><code> )</code>

<code>NumberOfDicksonNearFields</code> 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.

<pre>
 gap> NumberOfDicksonNearFields( 5, 4 );
 2
</pre>


<h2><a name="SECT003">10.3 Exceptional nearfields</a></h2>

<a name = "SSEC003.1"></a>
<li><code>ExceptionalNearFields( </code><var>q</var><code> )</code>

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.)

<code>ExceptionalNearFields</code> returns the list of exceptional nearfields for a given
size <var>q</var>.

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

<a name = "SSEC003.2"></a>
<li><code>AllExceptionalNearFields()</code>

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.)

<code>AllExceptionalNearFields</code> without argument returns the list of exceptional
nearfields.

<pre>
 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 ) ]
</pre>


<h2><a name="SECT004">10.4 Planar nearrings</a></h2>

<a name = "SSEC004.1"></a>
<li><code>PlanarNearRing( </code><var>G</var><code>, </code><var>phi</var><code>, </code><var>reps</var><code> )</code>

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, <a href="biblio.htm#Clay:Nearrings"><[>Clay:Nearrings</cite></a>:]
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.

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

<pre>
 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 )
</pre>

<a name = "SSEC004.2"></a>
<li><code>OrbitRepresentativesForPlanarNearRing( </code><var>G</var><code>, </code><var>phi</var><code>, </code><var>i</var><code> )</code>

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 <code>PlanarNearRing</code>) 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 <code>OrbitRepresentativesForPlanarNearRing</code>
returns precisely one set of representatives of cardinality <var>i</var> for each
isomorphism class of planar nearrings which can be generated from the
Ferrero pair ( <var>G</var>, <var>phi</var> ).

<pre>
 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
</pre>


<h2><a name="SECT005">10.5 Weakly divisible nearrings</a></h2>

<a name = "SSEC005.1"></a>
<li><code>WdNearRing( </code><var>G</var><code>, </code><var>psi</var><code>, </code><var>phi</var><code>, </code><var>reps</var><code> )</code>

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
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> eφψi·x : = xφψi </td></tr></table></td></tr></table>
Then (N,+,·) is a zerosymmetric (left) wd nearring.

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

<pre>
 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 )
</pre>


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