The original construction of Hall planes was based on the Hall quasifield (also called a Hall system), H of order for p a prime. The creation of the plane from the quasifield follows the standard construction (see quasifield for details).
To build a Hall quasifield, start with a Galois field, for p a prime and a quadratic irreducible polynomial over F. Extend , a two-dimensional vector space over F, to a quasifield by defining a multiplication on the vectors by when and otherwise.
Writing the elements of H in terms of a basis <1, λ>, that is, identifying (x,y) with x + λy as x and y vary over F, we can identify the elements of F as the ordered pairs (x, 0), i.e. x + λ0. The properties of the defined multiplication which turn the right vector space H into a quasifield are:
every element α of H not in F satisfies the quadratic equation f(α) = 0;
F is in the kernel of H (meaning that (α + β)c = αc + βc, and (αβ)c = α(βc) for all α, β in H and all c in F); and
every element of F commutes (multiplicatively) with all the elements of H.
Another construction that produces Hall planes is obtained by applying derivation to Desarguesian planes.
A process, due to T. G. Ostrom, which replaces certain sets of lines in a projective plane by alternate sets in such a way that the new structure is still a projective plane is called derivation. We give the details of this process. Start with a projective plane of order and designate one line as its line at infinity. Let A be the affine plane. A set D of points of is called a derivation set if for every pair of distinct points X and Y of A which determine a line meeting in a point of D, there is a Baer subplane containing X, Y and D (we say that such Baer subplanes belong to D.) Define a new affine plane as follows: The points of are the points of A. The lines of are the lines of which do not meet at a point of D (restricted to A) and the Baer subplanes that belong to D (restricted to A). The set is an affine plane of order and it, or its projective completion, is called a derived plane.
While usually constructed in the same way as other Hall planes, the Hall plane of order 9 was actually found earlier by Oswald Veblen and Joseph Wedderburn in 1907. There are four quasifields of order nine which can be used to construct the Hall plane of order nine. Three of these are Hall systems generated by the irreducible polynomials , or . The first of these produces an associative quasifield, that is, a near-field, and it was in this context that the plane was discovered by Veblen and Wedderburn. This plane is often referred to as the nearfield plane of order nine.
The Hall plane of order 9 admits four inequivalent embedded unitals. Two of these unitals arise from Buekenhout's constructions: one is parabolic, meeting the translation line in a single point, while the other is hyperbolic, meeting the translation line in 4 points. The latter of these two unitals was shown by Grüning to also be embeddable in the dual Hall plane. Another of the unitals arises from the construction of Barlotti and Lunardon. The fourth has an automorphism group of order 8 isomorphic to the quaternions, and is not part of any known infinite family.
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^Grüning, Klaus (1987-06-01). "A class of unitals of order which can be embedded in two different planes of order ". Journal of Geometry. 29 (1): 61–77. doi:10.1007/BF01234988. ISSN 1420-8997.
^Barlotti, A.; Lunardon, G. (1979). "Una classe di unitals nei -piani". Rivisita di Matematica della Università di Parma. 4: 781–785.
Dembowski, P. (1968), Finite Geometries, Berlin: Springer-Verlag