Near-field (mathematics)

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In mathematics a near-field is a structure, very much like a division ring, except for one axiom : the right distributive law.

[edit] Definition

A near-field (Q, + ,.) is a structure, where + and . are binary operations (the respective addition and multiplication) on Q, satisfying these axioms :

  • (Q, + ) is an abelian group with identity element 0.
  • (Q0,.) is a group
  • a.(b+c)=a.b+a.c \ \forall a,b,c\in Q

One can prove that the near-fields are just the quasifields with an associative multiplication.

[edit] Examples

We construct a near-field that is not a division ring of nine elements. Suppose K=GF(9). Let + and . denote the addition and multiplication respectively. We now define a new multiplication \otimes on the same set K :

u\otimes v=u.v if u is square in the original field
u\otimes v=u.v^3 if u is not square in the original field

One can check that (K,+,\otimes) is a near-field but not a division ring.

This near-field allows the construction of a projective plane that is not Desarguesian: the Hall plane.


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