In 1905, in the course of proving the independence of the field postulates, L. E. Dickson [Dic05a] (p. 203) introduced the first example of a nearfield. His example is a set of 9 elements with operations of addition and multiplication which satisfy all the axioms of a field except for the commutative law of multiplication and the right distributive law. Later that year Dickson [Dic05b] published a more extensive collection of examples: an infinite series obtained by twisting the multiplication of a Galois field and seven "irregular" examples.
The terminology `nearfield' seems to have introduced by Zassenhaus in his 1935 paper [Zas35] where he showed that the only finite nearfields (endliche Fastkörper) are those due to Dickson.
The irregular nearfields are often referred to as Zassenhaus nearfields and the nearfields in the infinite series are called Dickson nearfields.
In the papers of Dickson and Zassenhaus the nearfields are left-distributive but for the purposes of the Magma implementation we consider only right-distributive nearfields.
Nearfields are important in group theory, geometry and a combination of these two fields. On the one hand, the finite sharply doubly transitive permutation groups are in one-to-one correspondence with the finite nearfields and on the other hand, nearfields coordinatise a class of translation planes [Hal59], [L"69] and they are the starting point for the construction of the Hughes planes [Dem71], [Hug57]. Furthermore, every sharply transitive collineation group of projective space over a finite field is a quotient of the group of units of a nearfield [EK63] (see also, [Dem68, S1.4, n()scriptstyle o 17]).