Related pagesThe affine plane over a vector space
Fundamental theorem of projective geometry
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
is an isomorphism of K onto the ternary field of PG(V) with respect to (o,e,u,v).