In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a near-field is an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
similar to a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
, except that it has only one of the two distributive laws. Alternatively, a near-field is a
near-ring In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Definition
A set ''N'' together with two binary operations ...
in which there is a
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
and every non-zero element has a
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
.
Definition
A near-field is a set
together with two
binary operations
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary o ...
,
(addition) and
(multiplication), satisfying the following axioms:
:A1:
is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
.
:A2:
=
for all elements
,
,
of
(The
associative law
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
for multiplication).
:A3:
for all elements
,
,
of
(The right
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
).
:A4:
contains an element 1 such that
for every element
of
(
Multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
).
:A5: For every non-zero element
of
there exists an element
such that
(
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
).
Notes on the definition
# The above is, strictly speaking, a definition of a ''right'' near-field. By replacing A3 by the left distributive law
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 In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A qu ...
. 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.
[H. Zassenhaus,]
Über endliche Fastkörper
in ''Abh. Math. Semin. Univ. Hambg.'' 11 (1935), 187-220. 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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
under multiplication.
#:However, this alternative definition includes one exceptional structure of order 2 which fails to satisfy various basic theorems (such as
for all
). 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
for all
and
.
Examples
# Any
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
(including any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
) 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
be the
Galois field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of order 9. Denote multiplication in
by '
'. Define a new binary operation ' · ' by:
#::If
is any element of
which is a square and
is any element of
then
.
#::If
is any element of
which is not a square and
is any element of
then
.
#:Then
is a near-field with this new multiplication and the same addition as before.
History and applications
The concept of a near-field was first introduced by
Leonard Dickson
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ...
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 incidence 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 In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4.
Algebraic cons ...
, 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 cipher
In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than t ...
s.
Description in terms of Frobenius groups and group automorphisms
Let
be a near field. Let
be its multiplicative group and let
be its additive group. Let
act on
by
. The axioms of a near field show that this is a right group action by group automorphisms of
, and the nonzero elements of
form a single orbit with trivial stabilizer.
Conversely, if
is an abelian group and
is a subgroup of
which acts freely and transitively on the nonzero elements of
, then we can define a near field with additive group
and multiplicative group
. Choose an element in
to call
and let
be the bijection
. Then we define addition on
by the additive group structure on
and define multiplication by
.
A
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius.
Structure
Suppos ...
can be defined as a finite group of the form
where
acts without stabilizer on the nonzero elements of
. Thus, near fields are in bijection with Frobenius groups where
.
Classification
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
where
is an abelian group and
is a group of automorphisms of
which acts freely and transitively on the nonzero elements of
.
The construction of Dickson proceeds as follows.
[M. Hall, 20.7.2, ''The Theory of Groups'', Macmillan, 1959] Let
be a prime power and choose a positive integer
such that all prime factors of
divide
and, if
, then
is not divisible by
. Let
be the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of order
and let
be the additive group of
. The multiplicative group of
, together with the
Frobenius automorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
generate a group of automorphisms of
of the form
, where
is the cyclic group of order
. The divisibility conditions on
allow us to find a subgroup of
of order
which acts freely and transitively on
. The case
is the case of commutative finite fields; the nine element example above is
,
.
In the seven exceptional examples,
is of the form
. 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 solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s; these rotational symmetry groups are
,
and
respectively.
and
can also be described as
and
.
See also
*
Near-ring In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Definition
A set ''N'' together with two binary operations ...
*
Planar ternary ring
In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marsh ...
*
Quasifield In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A qu ...
References
{{reflist
External links
Nearfieldsby Hauke Klein.
Algebraic structures
Projective geometry