Wild nearfields of dimension 2 over rational function fields

Abstract

Let F be a field. We construct many subgroups of GL2(F (t)) that act regularly on F (t) 2 \{0}, and we show that the corresponding nearfields are not Dickson nearfields if char F = 2.Mathematics Subject Classification. 12K05, 20H20, 20E06.Keywords. Nearfield, Dickson nearfield, rational function field.Nearfields are generalisations of skew fields: just drop one of the two distributive laws. Many examples can be obtained by distorting the multiplication of a skew field D using automorphisms of D, as explained in Sect. 2. These examples are called Dickson nearfields; they are tame in the sense that they are closely related to skew fields. All other nearfields we call wild. A famous result of Zassenhaus [15] says that there exist only seven wild finite nearfields.With hindsight, the first examples of infinite wild nearfields have been constructed already by Neumann [9], see [3, Section 1]. These nearfields are 2-dimensional vector spaces over Q. A kind of free construction of infinite wild nearfields is due to Zassenhaus [16]; see also [3,4]. This construction produces nearfields that are vector spaces over purely transcendental field extensions of infinite transcendency degree.As observed in [2], for p ∈ {2, 3} the commutator group PSL 2 (F p [t]) acts regularly (i.e. sharply transitively) on the projective line F p (t)∪{∞}, and this leads to wild nearfields of dimension 2 over the rational function field F p (t). Here we construct for many fields F a plethora of (non-normal) subgroups of SL 2 (F [t]) which lead to wild infinite nearfields of dimension 2 over the rational function field F (t); see Theorems 1.2, 1.5, 3.1 and 3.2. This construction uses suitable actions of SL 2 (F ) and the decomposition of SL 2 (F [t]) as an amalgamated free product according to Nagao [8].