Five wild nearfields

Abstract

Abstract. The projective groups PSL 2 .F p OEt / with p 2 ¹2; 3º and three related groups have normal subgroups that are sharply transitive on the corresponding projective line. This leads to five infinite nearfields which are not Dickson nearfields.

Wild nearfields of dimension 2 over rational function fields

“…The nearfield N is wild by [4,Theorem 1]; see also the proof of Proposition 3.2 in [2]. In particular, N is not a skew field, hence F (t) is the kernel of N .…”

“…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).…”

“…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.…”

“…For example, for F ∈ {F 2 , F 3 } we could take SL 2 (F ) as a point stabiliser; then X = SL 2 (F [t]) , and this leads to two of the nearfields constructed in [2].…”

“…The group X is transitive on F (t) ∪{∞}, hence irreducible on the vector space V := F (t) 2 . The centraliser C of X in the endomorphism ring of V is a skew field by Schur's Lemma, and F (t) id is contained in the center of C. We have dim F (t) id C ≤ 2, because the evaluation map C → V : c → c(v) is linear and injective for every non-zero vector v ∈ V .…”