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{{About|the mathematical concept|the electromagnetic concept|Near and far field}}

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 <math>Q</math>, together with two [[binary operations]], <math>+</math> (addition) and <math>\cdot</math> (multiplication), satisfying the following axioms:
:A1: <math>(Q, +)</math> is an [[Abelian group]].
:A2: <math>(a \cdot b) \cdot c</math> = <math>a \cdot (b \cdot c)</math> for all elements <math>a</math>, <math>b</math>, <math>c</math> of <math>Q</math> (The [[associative law]] for multiplication).
:A3: <math>(a + b) \cdot c = a \cdot c + b \cdot c</math> for all elements <math>a</math>, <math>b</math>, <math>c</math> of <math>Q</math> (The right [[distributive law]]).
:A4: <math>Q</math> contains an element 1 such that <math>1 \cdot a = a \cdot 1 = a</math> for every element <math>a</math> of <math>Q</math> ([[Multiplicative identity]]).
:A5: For every non-zero element a of <math>Q</math> there exists an element <math>a^{-1}</math> such that <math> a \cdot a^{-1} = 1 = a^{-1} \cdot a </math> ([[Multiplicative inverse]]).

=== Notes on the definition ===
# The above is strictly a definition of a ''right'' near-field. By replacing A3 by the left distributive law <math> c \cdot (a + b) = c \cdot a + c\cdot b </math> 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 near-field can be equivalently defined as a right [[quasifield]] with associative multiplication.
# 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.<ref>J.L. Zemmer, "The additive group of an infinite near-field is abelian" in ''J. London Math. Soc.'' 44 (1969), 65-67.</ref><ref name="Zassenhaus">H Zassenhaus, Abh. Math. Sem. Hans. Univ. 11, pp 187-220.</ref><ref>B.H. Neumann, "On the commutativity of addition" in ''J. London Math. Soc.'' 15 (1940), 203-208.</ref> 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:
#:A4*: The non-zero elements form a [[Group (mathematics)|group]] under multiplication.
#:However, this alternative definition includes one exceptional structure of order 2 which fails to satisfy various basic theorems (such as <math> x \cdot 0 = 0</math> for all <math> x </math>). Thus it is much more convenient, and more usual, to use the axioms in the form given above. The difference is that A4 requires 1 to be an identity for all elements, A4* only for non-zero elements.
#:The exceptional structure can be defined by taking an additive group of order 2, and defining multiplication by <math> x \cdot y = x </math> for all <math> x </math> and <math> y</math>.

== Examples ==
# Any [[division ring]] (including any [[Field (mathematics)|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.
#:Let <math>K</math> be the [[Finite field|Galois field]] of order 9. Denote multiplication in <math>K</math> by ' <math>*</math> '. Define a new binary operation ' '''·''' ' by:
#::If <math> b </math> is any element of <math>K</math> which is a square and <math> a </math> is any element of <math>K</math> then <math> a \cdot b = a*b</math>.
#::If <math> b </math> is any element of <math>K</math> which is not a square and <math> a </math> is any element of <math>K</math> then <math> a \cdot b = a^3*b</math>.
#:Then <math>K</math> is a near-field with this new multiplication and the same addition as before.<ref>G. Pilz, Near-Rings, page 257.</ref>

== 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.<ref name="Zassenhaus"/>

The earliest application of the concept of near-field was in the study of geometries, such as [[projective geometry|projective geometries]].<ref>O. Veblen and J. H. Wedderburn "Non-desarguesian and non-pascalian geometrie" in ''Trans. Amer. Math. Soc.'' 8 (1907), 379-388.</ref><ref>P. Dembrowski "Finite geometries" Springer, Berlin, (1968).</ref> 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 (mathematician)|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.<ref>[[T. G. Room]] & P.B. Kirkpatrick (1971) ''Miniquaternion geometry'', §1.3 The Miniquaternion system <math>\mathcal(Q),</math>pp 8–20, [[Cambridge University Press]] ISBN 0-521-07926-8</ref>

There are numerous other applications, mostly to geometry.<ref>H. Wähling "Theorie der Fastkörper", Thales Verlag, Essen, (1987).</ref> A more recent application of near-fields is in the construction of ciphers for data-encryption, such as [[Hill cipher]]s.<ref>M. Farag, "Hill Ciphers over Near-Fields" in ''Mathematics and Computer Education'' v41 n1 (2007) 46-54.</ref>

== Description in terms of Frobenius groups and group automorphisms ==

Let <math>K</math> be a near field. Let <math>K_m</math> be its multiplicative group and let <math>K_a</math> be its additive group. Let <math>c \in K_m</math> act on <math>b \in K_a</math> by <math>b \mapsto b \cdot c</math>. The axioms of a near field show that this is a right group action by group automorphisms of <math>K_a</math>, and the nonzero elements of <math>K_a</math> form a single orbit with trivial stabilizer.

Conversely, if <math>A</math> is an abelian group and <math>M</math> is a subgroup of <math>\mathrm{Aut}(A)</math> which acts freely and transitively on the nonzero elements of <math>A</math>, then we can define a near field with additive group <math>A</math> and multiplicative group <math>M</math>. Choose an element in <math>A</math> to call <math>1</math> and let <math>\phi: M \to A \setminus \{ 0 \}</math> be the bijection <math>m \mapsto 1 \ast m</math>. Then we define addition on <math>A</math> by the additive group structure on <math>A</math> and define multiplication by <math>a \cdot b = 1 \ast \phi^{-1}(a) \phi^{-1}(b)</math>.

A [[Frobenius group]] can be defined as a finite group of the form <math>A \rtimes M</math> where <math>M</math> acts without stabilizer on the nonzero elements of <math>A</math>. Thus, near fields are in bijection with Frobenius groups where <math>|M| = |A|-1</math>.

== 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 <math>(A,M)</math> where <math>A</math> is an abelian group and <math>M</math> is a group of automorphisms of <math>A</math> which acts freely and transitively on the nonzero elements of <math>A</math>.

The construction of Dickson proceeds as follows.<ref>M. Hall, 20.7.2, ''The Theory of Groups'', Macmillian, 1959</ref> Let <math>q</math> be a prime power and choose a positive integer <math>n</math> such that all prime factors of <math>n</math> divide <math>q-1</math> and, if <math>q \equiv 3 \bmod 4</math>, then <math>n</math> is not divisible by <math>4</math>. Let <math>F</math> be the [[finite field]] of order <math>q^n</math> and let <math>A</math> be the additive group of <math>F</math>. The multiplicative group of <math>F</math>, together with the [[Finite field#Frobenius automorphisms|Frobenius automorphism]] <math>x \mapsto x^q</math> generate a group of automorphisms of <math>F</math> of the form <math>C_n \ltimes C_{q^n-1}</math>, where <math>C_k</math> is the cyclic group of order <math>k</math>. The divisibility conditions on <math>n</math> allow us to find a subgroup of <math>C_n \ltimes C_{q^n-1}</math> of order <math>q^n-1</math> which acts freely and transitively on <math>A</math>. The case <math>n=1</math> is the case of commutative finite fields; the nine element example above is <math>q=3</math>, <math>n=2</math>.

In the seven exceptional examples, <math>A</math> is of the form <math>C_p^2</math>. This table, including the numbering by Roman numerals, is taken from Zassenhaus's paper.<ref name="Zassenhaus"/>

{| class="wikitable"
|-
! !! <math>A=C_p^2</math> !! Generators for <math>M</math> !! Description(s) of <math>M</math>
|-
| I || <math>p=5</math> || <math>\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 1 & -2 \\ -1 & -2 \\ \end{smallmatrix} \right)</math> || <math>2T</math>, the [[binary tetrahedral group]].
|-
| II || <math>p=11</math> || <math>\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 1 & 5 \\ -5 & -2 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 4 & 0 \\ 0 & 4 \\ \end{smallmatrix} \right)</math> ||<math>2 T \times C_5</math>
|-
| III || <math>p=7</math> ||<math>\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 1 & 3 \\ -1 & -2 \\ \end{smallmatrix} \right)</math> || <math>2 O</math>, the [[binary octahedral group]].
|-
| IV || <math>p=23</math> || <math>\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 1 & -6 \\ 12 & -2 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 2 & 0 \\ 0 & 2 \\ \end{smallmatrix} \right)</math> || <math>2 O \times C_{11}</math>
|-
| V || <math>p=11</math> || <math>\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 2 & 4 \\ 1 & -3 \\ \end{smallmatrix} \right)</math> || <math>2 I</math>, the [[binary icosahedral group]].
|-
| VI || <math>p=29</math> || <math>\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 1 & -7 \\ -12 & -2 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 16 & 0 \\ 0 & 16 \\ \end{smallmatrix} \right)</math> || <math>2 I \times C_{7}</math>
|-
| VII || <math>p=59</math> || <math>\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 9 & 15 \\ -10 & -10 \\ \end{smallmatrix} \right)</math> <math>\left( \begin{smallmatrix} 4 & 0 \\ 0 & 4 \\ \end{smallmatrix} \right)</math> || <math>2 I \times C_{29}</math>
|}

The binary tetrahedral, octahedral and icosahedral groups are central extensions of the rotational symmetry groups of the [[platonic solid]]s; these rotational symmetry groups can are <math>A_4</math>, <math>S_4</math> and <math>A_5</math> respectively. <math>2T</math> and <math>2 I</math> can also be described as <math>SL(2,\mathbb{F}_3)</math> and <math>SL(2,\mathbb{F}_5)</math>.

== See also ==
* [[Nearring|Near-ring]]
* [[Planar ternary ring]]
* [[Quasifield]]

== References ==
{{reflist}}

==External links==
* [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/nearfield.html Nearfields] by Hauke Klein.

[[Category:Algebraic structures]]
[[Category:Projective geometry]]

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en>Monkbot: /* References */Task 5: Fix CS1 deprecated coauthor parameter errors

en>RjwilmsiBot: /* References */Fix cite template param names, coathor -> coauthor, using AWB (7701)