Quasifield

Strictly speaking, this is the definition of a left quasifield. A right quasifield is similarly defined, but satisfies right distributivity instead. A quasifield satisfying both distributive laws is called a semifield, in the sense in which the term is used in projective geometry.

The kernel K of a quasifield Q is the set of all elements c such that :

As a finite division ring is a finite field by Wedderburn's theorem, the order of the kernel of a finite quasifield is a prime power. The vector space construction implies that the order of any finite quasifield must also be a prime power.

A (right) near-field that is a (right) quasifield is called a "planar near-field".

The smallest quasifields are abelian and unique. They are the finite fields of orders up to and including eight. The smallest quasifields which are not division rings are the four non-abelian quasifields of order nine; they are presented in Hall, Jr. (1959) and Weibel (2007).

The plane does not uniquely determine the ring; all 4 nonabelian quasifields of order 9 are ternary rings for the unique non-Desarguesian translation plane of order 9. These differ in the fundamental quadrilateral used to construct the plane (see Weibel 2007).

Quasifields were called "Veblen-Wedderburn systems" in the literature before 1975, since they were first studied in the 1907 paper (Veblen-Wedderburn 1907) by O. Veblen and J. Wedderburn. Surveys of quasifields and their applications to projective planes may be found in Hall (1959) and Weibel (2007).