To build a Hall quasifield, start with a Galois field, F = GF(pn) for p a prime and a quadratic irreducible polynomial f(x) = x2 − 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 (a, b) ∘ (c, d) = (ac − bd−1f(c), ad − bc + br) when d ≠ 0 and (a, b) ∘ (c, 0) = (ac, bc) otherwise.
[8] The first of these produces an associative quasifield,[9] that is, a near-field, and it was in this context that the plane was discovered by Veblen and Wedderburn.
[11] The Hall plane of order 9 admits four inequivalent embedded unitals.
The latter of these two unitals was shown by Grüning[14] to also be embeddable in the dual Hall plane.
[15] The fourth has an automorphism group of order 8 isomorphic to the quaternions, and is not part of any known infinite family.