Information AboutQuasifield |
| CATEGORIES ABOUT QUASIFIELD | |
| nonassociative algebra | |
|
DEFINITION A quasifield (Q,+,.) is a structure, where + and . binary operations on Q, satisfying these axioms : Sometimes right distributivity is assumed instead. Although not assumed, one can prove that the axioms imply that the additive group is abelian. KERNEL The kernel K of a quasifield Q is the set of all elements k such that : Restricting the binary operations + and . to K, one can shown that (K,+,.) is a Division Ring . One can now make a vector space of Q over K, with the following scalar multiplication : As the order of any finite division ring is a Prime Power , this means that the order of any quasifield is also a prime power. PROJECTIVE PLANES We define a ternary map T from Q to Q: One can verify now that (Q,T) satisfies the axioms of a Planar Ternary Ring . Thus, given a quasifield, one can always construct a Projective Plane . If it is not a division ring, the plane is never desarguesian. However, as the order of a quasifield is always a prime power, one cannot expect to find counterexamples for the conjecture that the order of a projective plane is always a prime power. Projective planes constructed in this way have a unique geometric property : they are the only translation planes. |
|
|