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.
(multiplication), satisfying the following axioms: 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.
Hans Zassenhaus proved that all but 7 finite near-fields are either fields or Dickson near-fields.
[2] The earliest application of the concept of near-field was in the study of incidence geometries such as projective geometries.
[5][6] 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.
Kirkpatrick provided an alternative development.
[7] There are numerous other applications, mostly to geometry.
[8] A more recent application of near-fields is in the construction of ciphers for data-encryption, such as Hill ciphers.
form a single orbit with trivial stabilizer.
which acts freely and transitively on the nonzero elements of
, then we can define a near field with additive group
acts without stabilizer on the nonzero elements of
Thus, near fields are in bijection with Frobenius groups where
As mentioned 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
which acts freely and transitively on the nonzero elements of
The construction of Dickson proceeds as follows.
be a prime power and choose a positive integer
be the finite field of order
generate a group of automorphisms of
is the cyclic group of order
allow us to find a subgroup of
which acts freely and transitively on
is the case of commutative finite fields; the nine element example above is
This table, including the numbering by Roman numerals, is taken from Zassenhaus's paper.
[2] The binary tetrahedral, octahedral and icosahedral groups are central extensions of the rotational symmetry groups of the platonic solids; these rotational symmetry groups are