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There are two types of finite nearfield: the regular nearfields
of Dickson and the irregular nearfields of Zassenhaus. In order
to accommodate both types Magma has a `virtual type'
Nfd and types NfdDck and NfdZss which inherit from Nfd.
In order to begin exploring Nfd types in Magma we need a way to
create instances of nearfields and their elements. As already mentioned
there is a large class of nearfields first described by L. E. Dickson
[Dic05a], [Dic05b] in 1905 and in this
section we describe how to construct them in Magma.
The nearfields resulting from this construction will be called
Dickson (or regular) nearfields.
If p is a prime and if the positive integers h and v satisfy
 - if r is a prime or 4 and if r divides v, then
r divides ph - 1
then (p, h, v) is a Dickson triple.
If we write q = ph, the condition above is equivalent to
- All prime factors of v divide q - 1 and q ≡ 3 mod 4 implies
v ≢ 0 mod 4.
We call (q, v) a Dickson pair.
The isomorphism type of a Dickson nearfield depends on the choice of
primitive element of the underlying Galois field. It has been shown by
Lüneburg [L"71] that if φ is the Euler phi-function and
g is the order of p modulo v, there are φ(v)/g isomorphism classes
of Dickson nearfields with the same Dickson triple (p, h, v).
The default nearfield will use the `standard' primitive element of the
field. The other variants with the same Dickson pair can be obtained by
providing an integer s coprime to v. Internally this is converted
to a suitable integer e coprime to qv - 1 such that s ≡ e mod v.
The list of Dickson pairs (q, v) for prime p, where hlo and hhi
are the lower and upper bounds on h and where vlo and vhi
are the lower and upper bounds on v.
The list of Dickson pairs (ph, v) for the prime p, where h1 and
v1 are upper bounds on h and v.
The list of Dickson triples (p, h, v) for the prime p, where
hb and vb are bounds on h and v.
For each Dickson pair (equivalently Dickson triple), there is at least one
Dickson nearfield.
> DicksonPairs(5,3,4,4,5);
[
[ 125, 4 ],
[ 625, 4 ]
]
> DicksonPairs(5,4,5);
[
[ 5, 1 ],
[ 5, 2 ],
[ 5, 4 ],
[ 25, 1 ],
[ 25, 2 ],
[ 25, 3 ],
[ 25, 4 ],
[ 125, 1 ],
[ 125, 2 ],
[ 125, 4 ],
[ 625, 1 ],
[ 625, 2 ],
[ 625, 3 ],
[ 625, 4 ]
]
> DicksonTriples(5,4,5);
[
[ 5, 1, 1 ],
[ 5, 1, 2 ],
[ 5, 1, 4 ],
[ 5, 2, 1 ],
[ 5, 2, 2 ],
[ 5, 2, 3 ],
[ 5, 2, 4 ],
[ 5, 3, 1 ],
[ 5, 3, 2 ],
[ 5, 3, 4 ],
[ 5, 4, 1 ],
[ 5, 4, 2 ],
[ 5, 4, 3 ],
[ 5, 4, 4 ]
]
The number of non-isomorphic nearfields with Dickson pair (q, v).
The number of variants of the Dickson nearfield N.
Representatives for the variant parameter of nearfields with Dickson pair
(q, v).
For each Dickson pair there can be several variants. The variant
representative can be used when constructing the corresponding
Dickson nearfield.
> NumberOfVariants(625,4);
2
> VariantRepresentatives(625,4);
[ 1, 3 ]
Variant: RngIntElt Default: 1
LargeMatrices: BoolElt Default: false
Create a Dickson nearfield from the Dickson pair (q, v). The Variant
parameter is an integer s which can be used to specify the choice of
primitive element (see the discussion following the intrinsic
DicksonTriples). The parameter
LargeMatrices is used only when the group of units of the nearfield
is requested. The default is to represent the group of units as a matrix
group defined over the kernel of the nearfield. But if LargeMatrices is
true, the matrices are defined over the prime field.
As indicated in the previous example, up to isomorphism, there are two
Dickson nearfields with Dickson pair (625, 4).
> D := DicksonNearfield(625,4);
> D3 := DicksonNearfield(625,4 : Variant := 3);
> D5 := DicksonNearfield(625,4 : Variant := 5);
> D eq D3;
false
> D3 eq D5;
false
> D eq D5;
true
> D;
Nearfield D of Dickson type defined by the pair (625, 4)
Order = 152587890625
It was shown by Zassenhaus [Zas35] that in addition to the
regular nearfields there are seven irregular nearfields. Zassenhaus
gave constructions but did not prove their uniqueness.
The proofs in [Zas35] are known to contain gaps. Perhaps the
most reliable account of the existence and uniqueness of the irregular
nearfields is the PhD thesis of Dancs-Groves [Gro74].
The seven finite nearfields which are not Dickson nearfields are the
Zassenhaus nearfields.
Zassenhaus nearfields can be distinguished from regular nearfields by the
fact that the multiplicative group of a finite nearfield N is metacyclic
if and only if N is regular.
As a consequence, a Zassenhaus nearfield cannot occur as a subfield of a
Dickson nearfield.
Creates the nth Zassenhaus nearfield.
The orders of the Zassenhaus nearfields are 5 2, 11 2, 7 2, 23 2,
11 2, 29 2 and 59 2.
> for n := 1 to 7 do ZassenhausNearfield(n); end for;
Irregular nearfield Z with Zassenhaus number 1
Order = 25
Irregular nearfield Z with Zassenhaus number 2
Order = 121
Irregular nearfield Z with Zassenhaus number 3
Order = 49
Irregular nearfield Z with Zassenhaus number 4
Order = 529
Irregular nearfield Z with Zassenhaus number 5
Order = 121
Irregular nearfield Z with Zassenhaus number 6
Order = 841
Irregular nearfield Z with Zassenhaus number 7
Order = 3481
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