Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring D—. The projective plane is then called desarguesian. A theorem of Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.
- Pappus's hexagon theorem holds in a desarguesian projective plane if and only if D is a field; it corresponds to the identity .
- Fano's theorem (which states a certain intersection does not happen) holds in if and only if D has characteristic ; it corresponds to the identity a+a=0.
Read more about this topic: Intersection Theorem
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