Related pagesHomologies and elations of P(K,T)
Theorem. Let K be a quasifield. Then the additive group (K,+) is abelian and
is a subskewfield of K, called the kernel of K. Moreover, using the multiplication in K, the quasifield K becomes a vector space over it's kernel.
Since a quasifield K is a cartesian group, we may think of K as a ternary field by using the ternary operation T(s,x,t)=sx+t. In particular, we have a projective plane P(K) over K. These planes are exactly the translation planes.
Theorem. A ternary field (K,T) is a quasifield if and only if the line at infinity of the projective plane P(K,T) is a translation axis.
The kernel of a quasifield also has a simple geometrical interpretation, which is easily derived from the explicit computation of the homologies and elations of P(K,T).
Theorem. Let K be a quasifield and write Then