## Related pagesThe affine plane over a vector spaceDesarguesian planes 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(K^{3}).
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 e_{1}, e_{2},
e_{3} of V with

such that

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

Version