Famous Examples
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
Famous quotes containing the words famous and/or examples:
“Satan, what ails you? Where’s the famous tongue?
Thou onetime Prince of Conversationists?”
—Robert Frost (1874–1963)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)