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Near-field (mathematics)
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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.

  • (Q_{0},.) is a group

  • a.(b+c)=a.b+a.c \ orall a,b,c\in Q


One can prove that the near-fields are just the Quasifield s with an Associative Multiplication .


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 plane.