# Near-field (mathematics)

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 $Q$, together with two binary operations, $+$ (addition) and $\cdot$ (multiplication), satisfying the following axioms:

A1: $(Q, +)$ is an Abelian group.
A2: $(a \cdot b) \cdot c$ = $a \cdot (b \cdot c)$ for all elements $a$, $b$, $c$ of $Q$ (The associative law for multiplication).
A3: $(a + b) \cdot c = a \cdot c + b \cdot c$ for all elements $a$, $b$, $c$ of $Q$ (The right distributive law).
A4: $Q$ contains an element 1 such that $1 \cdot a = a \cdot 1 = a$ for every element $a$ of $Q$ (Multiplicative identity).
A5: For every non-zero element a of $Q$ there exists an element $a^{-1}$ such that $a \cdot a^{-1} = 1 = a^{-1} \cdot a$ (Multiplicative inverse).

### Notes on the definition

1. The above is strictly a definition of a right near-field. By replacing A3 by the left distributive law $c \cdot (a + b) = c \cdot a + c\cdot b$ 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.
2. A near-field can be equivalently defined as a right quasifield with associative multiplication.
3. 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.[1][2][3] 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.
4. 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 under multiplication.
However, this alternative definition includes one exceptional structure of order 2 which fails to satisfy various basic theorems (such as $x \cdot 0 = 0$ for all $x$). Thus it is much more convenient 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 $x \cdot y = x$ for all $x$ and $y$.
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## Examples

1. Any division ring (including any field) is a near-field.
2. The following defines a (right) near-field of order 9. It is the smallest near-field which is not a field.
Let $K$ be the Galois field of order 9. Denote multiplication in $K$ by ' $*$ '. Define a new binary operation ' · ' by:
If $b$ is any element of $K$ which is a square and $a$ is any element of $K$ then $a \cdot b = a*b$.
If $b$ is any element of $K$ which is not a square and $a$ is any element of $K$ then $a \cdot b = a^3*b$.
Then $K$ is a near-field with this new multiplication and the same addition as before.[4]
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## 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 division rings or Dickson near-fields.[2]

The earliest application of the concept of near-field was in the study of 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'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.[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.[9]

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## References

1. ^ J.L. Zemmer, "The additive group of an infinite near-field is abelian" in J. London Math. Soc. 44 (1969), 65-67.
2. ^ a b H Zassenhaus, Abh. Math. Sem. Hans. Univ. 11, pp 187-220.
3. ^ B.H. Neumann, "On the commutativity of addition" in J. London Math. Soc. 15 (1940), 203-208.
4. ^ G. Pilz, Near-Rings, page 257.
5. ^ O. Veblen and J. H. Wedderburn "Non-desarguesian and non-pascalian geometrie" in Trans. Amer. Math. Soc. 8 (1907), 379-388.
6. ^ P. Dembrowski "Finite geometries" Springer, Berlin, (1968).
7. ^ T. G. Room & P.B. Kirkpatrick (1971) Miniquaternion geometry, §1.3 The Miniquaternion system $\mathcal(Q),$pp 8–20, Cambridge University Press ISBN 0-521-07926-8
8. ^ H. Wähling "Theorie der Fastkörper", Thales Verlag, Essen, (1987).
9. ^ M. Farag, "Hill Ciphers over Near-Fields" in Mathematics and Computer Education v41 n1 (2007) 46-54.
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