Hall plane

In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order p²ⁿ for every prime p and every positive integer n provided p²ⁿ > 4.

The original construction of Hall planes was based on a Hall quasifield (also called a Hall system), H of order p²ⁿ for p a prime. The construction of the plane is the standard construction based on a quasifield (see Quasifield#Projective planes for the details.).

To build a Hall quasifield, start with a Galois field, ${\displaystyle F=GF(p^{n})}$ for p a prime and a quadratic irreducible polynomial ${\displaystyle f(x)=x^{2}-rx-s}$ over F. Extend H = F × F, a two-dimensional vector space over F, to a quasifield by defining a multiplication on the vectors by ${\displaystyle (a,b)\circ (c,d)=(ac-bd^{-1}f(c),ad-bc+br)}$ when ${\displaystyle d\neq 0}$ and ${\displaystyle (a,b)\circ (c,0)=(ac,bc)}$ otherwise.

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