The projective plane over a vector space

Related pages

The affine plane over a vector space
Desarguesian planes
Fundamental theorem of projective geometry
Let V be a three dimensional vector space over a skewfield K. We define a projective plane PG(V) as follows. The points of PG(V) are the one dimensional subspaces of V, and the lines are the two dimensional subspaces of V, respectively. A point p is incident with a line l if p is a subset of l.

A projective plane is isomorphic to some PG(V) if and only if is desarguesian. Moreover, given two three dimensional vector spaces V and V' over skewfields K and K', respectively, then the projective planes PG(V) and PG(V') are isomorphic if and only if the skewfields K and K' are isomorphic.

We may think of the skewfield K as a ternary field by using the ternary operation T(s,x,t)=sx+t. Then the projective plane over the ternary field K is isomorphic to the projective plane PG(K3). For example, an isomorphism is given by

In fact, all coordinatizing ternary fields of PG(V) are isomorphic with K.

Theorem. Let V be a three dimensional vector space over a skewfield K. Let (o,e,u,v) be a quadrangle in the projective plane PG(V). Then there exists a basis e1, e2, e3 of V with

u=e1K, v=e2K, o=e3K and e=(e1+e2+e3)K

such that

is an isomorphism of K onto the ternary field of PG(V) with respect to (o,e,u,v).

Contributed by Hauke Klein
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