The affine plane over a vector space

Related pages

The projective plane over a vector space
Desarguesian planes
Fundamental theorem of affine geometry
Let V be a two dimensional vector space over a skewfield K. We define an affine plane AG(V) as follows. The points of AG(V) are the elements of V, and the lines are the subsets of the form l=t+uK with a vector u distinct from zero.

An affine plane is isomorphic to some AG(V) if and only if the projective closure of is desarguesian. Moreover, given two vector space V, V' of dimension two over skewfields K and K', respectively, then the affine planes AG(V) and AG(V') are isomorphic if and only if the skewfields K and K' are isomorphic.

Given a vector space V of dimension 2 over a skewfield K, then

via the isomorphism

Contributed by Hauke Klein
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