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.
Contents
Definition
A near-field is a set
Notes on the definition
- The above is strictly a definition of a right near-field. By replacing A3 by the left distributive law
c ⋅ ( a + b ) = c ⋅ a + c ⋅ b we get a left near-field instead. Most commonly, "near-field" is taken as meaning "right near-field", but this is not a universal convention. - A (right) near-field is called "planar" if it is also a right quasifield. Every finite near-field is planar, but infinite near-fields need not be.
- It is not necessary to specify that the additive group is abelian, as this follows from the other axioms, as proved by B.H. Neumann and J.L. Zemmer. However, the proof is quite difficult, and it is more convenient to include this in the axioms so that progress with establishing the properties of near-fields can start more rapidly.
- Sometimes a list of axioms is given in which A4 and A5 are replaced by the following single statement:
Examples
- Any division ring (including any field) is a near-field.
- The following defines a (right) near-field of order 9. It is the smallest near-field which is not a field.
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't. 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
Conversely, if
A Frobenius group can be defined as a finite group of the form
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
The construction of Dickson proceeds as follows. Let
In the seven exceptional examples,
The binary tetrahedral, octahedral and icosahedral groups are central extensions of the rotational symmetry groups of the platonic solids; these rotational symmetry groups are