Editing Projective plane (section)

=== {{anchor|Desarguesian}} Desargues' theorem and Desarguesian planes ===

The [[Desargues' theorem|theorem of Desargues]] is universally valid in a projective plane if and only if the plane can be constructed from a three-dimensional vector space over a skewfield as [[#Vector space construction|above]].<ref>[[David Hilbert]] proved the more difficult "only if" part of this result.</ref> These planes are called '''Desarguesian planes''', named after [[Girard Desargues]]. The real (or complex) projective plane and the projective plane of order 3 given [[Projective plane#Some examples|above]] are examples of Desarguesian projective planes. The projective planes that can not be constructed in this manner are called [[non-Desarguesian plane]]s, and the [[Moulton plane]] given [[Projective plane#Some examples|above]] is an example of one. The PG(2,&nbsp;''K'') notation is reserved for the Desarguesian planes. When ''K'' is a [[Field (mathematics)|field]], a very common case, they are also known as ''field planes'' and if the field is a [[finite field]] they can be called [[Galois geometry|''Galois planes'']].