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| {{About|the mathematical concept|the electromagnetic concept|Near and far field}}
| | Wilber Berryhill is what his spouse enjoys to call him and he totally enjoys this title. Ohio is where his home is and his family members loves it. My husband doesn't like it the way I do but what I truly like doing is caving but I don't have the time recently. My working day job is a journey agent.<br><br>my web-site ... are psychics real, [http://www.youronlinepublishers.com/authWiki/AdolphvhBladenqq www.youronlinepublishers.com], |
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| 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]].
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| == Definition ==
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| 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:
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| :A1: <math>(Q, +)</math> is an [[Abelian group]].
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| :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).
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| :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]]).
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| :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]]).
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| :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]]).
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| === Notes on the definition ===
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| # 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.
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| # A near-field can be equivalently defined as a right [[quasifield]] with associative multiplication.
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| # 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.
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| # Sometimes a list of axioms is given in which A4 and A5 are replaced by the following single statement:
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| #:A4*: The non-zero elements form a [[Group (mathematics)|group]] under multiplication.
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| #: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.
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| #: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>.
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| == Examples ==
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| # Any [[division ring]] (including any [[Field (mathematics)|field]]) is a near-field.
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| # The following defines a (right) near-field of order 9. It is the smallest near-field which is not a field.
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| #: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:
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| #::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>.
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| #::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>.
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| #: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>
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| == History and Applications ==
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| 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.
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| [[Hans Zassenhaus]] proved that all but 7 finite near-fields are either fields or Dickson near-fields.<ref name="Zassenhaus"/>
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| 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>
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| 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>
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| == Description in terms of Frobenius groups and group automorphisms ==
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| 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.
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| 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>.
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| 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>.
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| == Classification ==
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| 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>.
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| 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>.
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| 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"/>
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| {| class="wikitable"
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| ! !! <math>A=C_p^2</math> !! Generators for <math>M</math> !! Description(s) of <math>M</math>
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| | 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]].
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| | 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>
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| | 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]].
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| | 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>
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| | 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]].
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| | 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>
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| | 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>
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| |}
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| 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>.
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| == See also ==
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| * [[Nearring|Near-ring]]
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| * [[Planar ternary ring]]
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| * [[Quasifield]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| * [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/nearfield.html Nearfields] by Hauke Klein.
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| [[Category:Algebraic structures]]
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| [[Category:Projective geometry]]
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Wilber Berryhill is what his spouse enjoys to call him and he totally enjoys this title. Ohio is where his home is and his family members loves it. My husband doesn't like it the way I do but what I truly like doing is caving but I don't have the time recently. My working day job is a journey agent.
my web-site ... are psychics real, www.youronlinepublishers.com,