Two Step Descent in Modular Galois

We propound a Descent Principle by which previously constructed equations over GF(q n)(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups, where q = p u > 1 is a power of a prime p and n is a positive integer. Currently this is achieved by starting with a vectorial (= additive) q-polynomial of q-degree m with Galois group GL(m; q) where m is any positive integer and then, under suitable conditions, enlarging its Galois group to GL(m; q n) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degree n. So, alternatively, we may regard this as an Ascent Principle. Elsewhere we proved this when m is square-free with GCD(mnu; 2p) = 1 = GCD(m; n). There the proof was based on CT (= the Classiication Theorem of Finite Simple Groups) in its incarnation of CPT (= the Classiication of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors). Here, without using CT, we shall give a direct proof of it when m = n = 2. The basic cornerstone of CPT is a Theorem of Burnside which says that a two-transitive permutation group has a unique minimal normal subgroup, which is either elementary abelian or simple. Since it is not very easy to nd a proof of this Theorem of Burnside in the modern literature, we shall include a self-contained proof of it. We shall also discuss the relationship of Burnside's Theorem to Cayley's Theorem, and apply it to Hilbert's Thirteenth Problem. Finally we shall motivate the whole matter by ideas from elementary analytic geometry. Section 1: Introduction Let q = p u > 1 be a power of a prime p, let m > 0 and n > 0 be integers, and let GF(q) k q K be elds where is an algebraic closure of K and where, as usual, GF(q) denotes the Galois eld of q elements. Also let E = E(Y) be a monic separable vectorial q-polynomial of q-degree m in Y over K, where the elements X 1 ; : : : ; X m need not be algebraically independent over k q. Now the set of all roots of E in is an m-dimensional GF(q)-vector-subspace V of and, since GF(q) is assumed to be a subbeld of k q and hence of K, every K-automorphism of the splitting eld K(V) of …