<html><head><title>[SONATA] 10 Nearfields, planar nearrings and weakly divisible nearrings</title></head>
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<h1>10 Nearfields, planar nearrings and weakly divisible nearrings</h1><p>
<P>
<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><p>
<p>
A <strong>nearfield</strong> is a nearring with <var>1</var> 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 citeWaehling:Fastkoerper.
<p>
Let <var>(N,+,cdot)</var> be a left nearring. For <var>a,b inN</var> we define <var>a equivb</var>
iff <var>acdotn = bcdotn</var> for all <var>ninN</var>. If <var>a equivb</var>, then <var>a</var> and <var>b</var>
are called <strong>equivalent multipliers</strong>.
A nearring <var>N</var> is called <strong>planar</strong> if <var>| N/<sub>equiv</sub> | ge3</var> and if
for any two non-equivalent multipliers <var>a</var> and <var>b</var> in <var>N</var>, for any <var>cinN</var>,
the equation <var>acdotx = bcdotx + c</var> has a unique solution.
See citeClay:Nearrings for basic results on planar nearrings.
<p>
All finite nearfields are planar nearrings.
<p>
A left nearring <var>(N,+,cdot)</var> is called <strong>weakly divisible</strong> if
<var>foralla,binN existsxinN : acdotx = b</var> or <var>bcdotx = a</var>.
<p>
All finite integral planar nearrings are weakly divisible.
<p>
<p>
<h2><a name="SECT001">10.1 Dickson numbers</a></h2>
<p><p>
<a name = "SECT001"></a>
<li><code>IsPairOfDicksonNumbers( </code><var>q</var><code>, </code><var>n</var><code> )</code>
<p>
A pair of Dickson numbers <var>(q,n)</var> consists of a prime power integer <var>q</var>
and a natural number <var>n</var> such that for <var>p = 4</var> or <var>p</var> prime, <var>p|n</var> implies
<var>p|q-1</var>.
<p>
<pre>
gap> IsPairOfDicksonNumbers( 5, 4 );
true
</pre>
<p>
<p>
<h2><a name="SECT002">10.2 Dickson nearfields</a></h2>
<p><p>
<a name = "SECT002"></a>
<li><code>DicksonNearFields( </code><var>q</var><code>, </code><var>n</var><code> )</code>
<p>
All finite nearfields with 7 exceptions can be obtained via socalled
coupling maps from finite fields. These nearfields are called Dickson
nearfields.
<p>
The multiplication map of such a Dickson nearfield is given by a pair of
Dickson numbers <var>(q,n)</var> in the following way:
<p>
Let <var>F = GF(q<sup>n</sup>)</var> and <var>w</var> be a primitive element of <var>F</var>. Let
<var>H</var> be the subgroup of <var>(Fsetminus{0},cdot)</var> generated by <var>w<sup>n</sup></var>.
Then <var>{w<sup>(q^i-1)/(q-1)</sup> | 0leqileqn-1 }</var> is a set of coset
representatives of <var>H</var> in <var>Fsetminus{0}</var>.
For <var>finHw<sup>(q^i-1)/(q-1)</sup></var> and <var>xinF</var> define <var>f*x = fcdotx<sup>q^i</sup></var>
and <var>0*x = 0</var>. Then <var>*</var> is a nearfield multiplication on the additive group
<var>(F,+)</var>.
<p>
Note that a Dickson nearfield is not uniquely determined by <var>(q,n)</var>, since
<var>w</var> can be chosen arbitrarily. Different choices of <var>w</var> may yield isomorphic
nearfields.
<p>
<code>DicksonNearFields</code> returns a list of the non-isomorphic Dickson nearfields
determined by the pair of Dickson numbers <var>(q,n)</var>
<p>
<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>
<p>
<a name = "SECT002"></a>
<li><code>NumberOfDicksonNearFields( </code><var>q</var><code>, </code><var>n</var><code> )</code>
<p>
<code>NumberOfDicksonNearFields</code> returns the number of non-isomorphic Dickson
nearfields which can be obtained from a pair of Dickson numbers <var>(q,n)</var>.
This number is given by <var>Phi(n)/k</var>. Here <var>Phi(n)</var> denotes the number
of relatively prime residues modulo <var>n</var> and <var>k</var> is the multiplicative order
of <var>p</var> modulo <var>n</var> where <var>p</var> is the prime divisor of <var>q</var>.
<p>
<pre>
gap> NumberOfDicksonNearFields( 5, 4 );
2
</pre>
<p>
<p>
<h2><a name="SECT003">10.3 Exceptional nearfields</a></h2>
<p><p>
<a name = "SECT003"></a>
<li><code>ExceptionalNearFields( </code><var>q</var><code> )</code>
<p>
There are 7 finite nearfields which cannot be obtained from finite fields
via a Dickson process. They are of size <var>p<sup>2</sup></var> for
<var>p = 5, 7, 11, 11, 23, 29, 59</var>. (There exist 2 exceptional nearfields of size
121.)
<p>
<code>ExceptionalNearFields</code> returns the list of exceptional nearfields for a given
size <var>q</var>.
<p>
<pre>
gap> ExceptionalNearFields( 25 );
[ ExplicitMultiplicationNearRing ( <pc group of size 25 with
2 generators> , multiplication ) ]
</pre>
<p>
<a name = "SECT003"></a>
<li><code>AllExceptionalNearFields()</code>
<p>
There are 7 finite nearfields which cannot be obtained from finite fields
via a Dickson process. They are of size <var>p<sup>2</sup></var> for
<var>p = 5, 7, 11, 11, 23, 29, 59</var>. (There exist 2 exceptional nearfields of size
121.)
<p>
<code>AllExceptionalNearFields</code> without argument returns the list of exceptional
nearfields.
<p>
<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>
<p>
<p>
<h2><a name="SECT004">10.4 Planar nearrings</a></h2>
<p><p>
<a name = "SECT004"></a>
<li><code>PlanarNearRing( </code><var>G</var><code>, </code><var>phi</var><code>, </code><var>reps</var><code> )</code>
<p>
A finite <strong>Ferrero pair</strong> is a pair of groups <var>(N,Phi)</var> where <var>Phi</var> is a
fixed-point-free automorphism group of <var>(N,+)</var>.
<p>
Starting with a Ferrero pair <var>(N,Phi)</var> we can construct a planar nearring
in the following way, citeClay:Nearrings:
Select representatives, say <var>e<sub>1</sub>,...,e<sub>t</sub></var>, for some or all of the
non-trivial orbits of <var>N</var> under <var>Phi</var>.
Let <var>C = Phi(e<sub>1</sub>)cup...cupPhi(e<sub>t</sub>)</var>.
For each <var>xinN</var> we define <var>a * x = 0</var> for <var>ainNsetminusC</var>, and
<var>a * x=phi<sub>a</sub>(x)</var> for <var>ainPhi(e<sub>i</sub>)subsetC</var> and <var>phi<sub>a</sub>(e<sub>i</sub>)=a</var>.
Then <var>(N,+,*)</var> is a (left) planar nearring.
<p>
Every finite planar nearring can be constructed from some Ferrero pair
together with a set of orbit representatives in this way.
<p>
<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>.
<p>
<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 generators> , multiplication )
</pre>
<p>
<a name = "SECT004"></a>
<li><code>OrbitRepresentativesForPlanarNearRing( </code><var>G</var><code>, </code><var>phi</var><code>, </code><var>i</var><code> )</code>
<p>
Let <var>(N,Phi)</var> be a Ferrero pair, and let <var>E = { e<sub>1</sub>,...,e<sub>s</sub> }</var> and
<var>F = { f<sub>1</sub>,...,f<sub>t</sub> }</var> be two sets of non-zero orbit representatives.
The nearring obtained from <var>N,Phi, E</var> by the Ferrero construction
(see <code>PlanarNearRing</code>) is isomorphic to the nearring obtained from <var>N,Phi, F</var>
iff there exists an automorphism <var>alpha</var> of <var>(N,+)</var> that normalizes <var>Phi</var>
such that
<var>{ alpha(e<sub>1</sub>),...,alpha(e<sub>s</sub>) } = { f<sub>1</sub>,...,f<sub>t</sub> }</var>.
<p>
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> ).
<p>
<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>
<p>
<p>
<h2><a name="SECT005">10.5 Weakly divisible nearrings</a></h2>
<p><p>
<a name = "SECT005"></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>
<p>
Every finite (left) weakly divisible nearring <var>(N,+,cdot)</var> can be constructed
in the following way:
<p>
(1) Let <var>psi</var> be an endomorphism of the group <var>(N,+)</var> such that Ker
<var>psi=</var> Image <var>psi<sup>r-1</sup></var> for some integer <var>r, r>0</var>. (Let <var>psi<sup>0</sup> :=</var> id.)
<p>
(2) Let <var>Phi</var> be an automorphism group of <var>(N,+)</var> such that
<var>psiPhisubseteqPhipsi</var> and <var>Phi</var> acts fixed-point-free on
<var>Nsetminus</var> Image <var>psi</var>.
(That is, for each
<var>varphiinPhi</var> there exists <var>varphi'inPhi</var> such that
<var>psivarphi= varphi'psi</var> and for all <var>ninNsetminus</var> Image <var>psi</var> the
equality <var>n^varphi= n</var> implies <var>varphi=</var> id. Note that our functions
operate from the right just like GAP-mappings do.)
<p>
(3) Let <var>EsubseteqN</var> be a complete set of orbit representatives for
<var>Phi</var> on <var>Nsetminus</var> Image <var>psi</var>, such that for all <var>e<sub>1</sub>, e<sub>2</sub>inE</var>, for all
<var>varphiinPhi</var> and for all <var>1 leqi leqr-1</var> the equality
<var>e<sub>1</sub><sup>varphipsi^i</sup> = e<sub>2</sub><sup>psi^i</sup></var> implies <var>varphipsi<sup>i</sup> = psi<sup>i</sup></var>.
<p>
Then for all <var>ninN, nneq0</var>, there are <var>igeq0 ,varphiinPhi</var> and
<var>einE</var> such that <var>n = e<sup>varphipsi^i</sup></var>; furthermore, for fixed <var>n</var>, the
endomorphism <var>varphipsi<sup>i</sup></var> is independent of the choice of <var>e</var> and
<var>varphi</var> in the representation of <var>n</var>.
<p>
For all <var>xinN, einE,varphiinPhi</var> and <var>igeq0</var> define <var>0cdotx := 0</var>
and
<p><var> e<sup>varphipsi^i</sup>cdotx := x<sup>varphipsi^i</sup> <p></var>
Then <var>(N,+,cdot)</var> is a zerosymmetric (left) wd nearring.
<p>
<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).
<p>
<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>
<p>
<p>
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<P>
<address>SONATA manual<br>November 2008
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