Descent principle in modular Galois theory

Abstract

We propound a descent principle by which previously constructed equations over GF.qn/.X/ may be deformed to have incarnations over GF.q/.X/ without changing their Galois groups. Currently this is achieved by starting with a vectorial (= additive) q-polynomial of q-degreemwith Galois group GL.m; q/ and then, under suitable conditions, enlarging its Galois group to GL.m; qn/ by forming its generalized iterate relative to an auxiliary irreducible polynomial of degree n. Elsewhere this was proved under certain conditions by using the classification of finite simple groups, and under some other conditions by using Kantor’s classification of linear groups containing a Singer cycle. Now under different conditions we prove it by using Cameron-Kantor’s classification of two-transitive linear groups.