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:
“The [Loyal] legion has taken the place of the club—the famous Cincinnati Literary Club—in my affections.... The military circles are interested in the same things with myself, and so we endure, if not enjoy, each other.”
—Rutherford Birchard Hayes (1822–1893)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)