Near-field (mathematics)

In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity, and every non-zero element has a multiplicative inverse.
Definition.
A near-field is a set formula_1, together with two binary operations, formula_2 (addition) and formula_3 (multiplication), satisfying the following axioms:
History and applications.
The concept of a near-field was first introduced by Leonard Dickson in 1905. He took division rings and modified their multiplication, while leaving addition as it was, and thus produced the first known examples of near-fields that were not division rings. The near-fields produced by this method are known as Dickson near-fields; the near-field of order 9 given above is a Dickson near-field.
Hans Zassenhaus proved that all but 7 finite near-fields are either fields or Dickson near-fields.
The earliest application of the concept of near-field was in the study of geometries, such as projective geometries. Many projective geometries can be defined in terms of a coordinate system over a division ring, but others can not. It was found that by allowing coordinates from any near-ring the range of geometries which could be coordinatized was extended. For example, Marshall Hall used the near-field of order 9 given above to produce a Hall plane, the first of a sequence of such planes based on Dickson near-fields of order the square of a prime. In 1971 T. G. Room and P.B. Kirkpatrick provided an alternative development.
There are numerous other applications, mostly to geometry. A more recent application of near-fields is in the construction of ciphers for data-encryption, such as Hill ciphers.
Description in terms of Frobenius groups and group automorphisms.
Let formula_29 be a near field. Let formula_44 be its multiplicative group and let formula_45 be its additive group. Let formula_46 act on formula_47 by formula_48. The axioms of a near field show that this is a right group action by group automorphisms of formula_45, and the nonzero elements of formula_45 form a single orbit with trivial stabilizer.
Conversely, if formula_51 is an abelian group and formula_52 is a subgroup of formula_53 which acts freely and transitively on the nonzero elements of formula_51, then we can define a near field with additive group formula_51 and multiplicative group formula_52. Choose an element in formula_51 to call formula_58 and let formula_59 be the bijection formula_60. Then we define addition on formula_51 by the additive group structure on formula_51 and define multiplication by formula_63.
A Frobenius group can be defined as a finite group of the form formula_64 where formula_52 acts without stabilizer on the nonzero elements of formula_51. Thus, near fields are in bijection with Frobenius groups where formula_67.
Classification.
As described above, Zassenhaus proved that all finite near fields either arise from a construction of Dickson or are one of seven exceptional examples. We will describe this classification by giving pairs formula_68 where formula_51 is an abelian group and formula_52 is a group of automorphisms of formula_51 which acts freely and transitively on the nonzero elements of formula_51.
The construction of Dickson proceeds as follows. Let formula_73 be a prime power and choose a positive integer formula_74 such that all prime factors of formula_74 divide formula_76 and, if formula_77, then formula_74 is not divisible by formula_79. Let formula_80 be the finite field of order formula_81 and let formula_51 be the additive group of formula_80. The multiplicative group of formula_80, together with the Frobenius automorphism formula_85 generate a group of automorphisms of formula_80 of the form formula_87, where formula_88 is the cyclic group of order formula_89. The divisibility conditions on formula_74 allow us to find a subgroup of formula_87 of order formula_92 which acts freely and transitively on formula_51. The case formula_94 is the case of commutative finite fields; the nine element example above is formula_95, formula_96.
In the seven exceptional examples, formula_51 is of the form formula_98. This table, including the numbering by Roman numerals, is taken from Zassenhaus's paper.
The binary tetrahedral, octahedral and icosahedral groups are central extensions of the rotational symmetry groups of the platonic solids; these rotational symmetry groups are formula_99, formula_100 and formula_101 respectively. formula_102 and formula_103 can also be described as formula_104 and formula_105.