On the Galois groups of iterated generic additive polynomials

Abstract

Let p > 0 be prime, and let K be any field of characteristic p. Let X and Y be independent (commuting) variables over K. A polynomial f(X) ∈ K[X] is said to be additive (over K) if and only if f(X+Y) = f(X)+f(Y) in K[X, Y]. It is elementary (and classical) that the additive f(X) over K are precisely the polynomials of the type Σ_j=0^da_jX^pj, where d [ges ] 0 and a₀, …, a_d∈K. Now let k [ges ] 2, and let X, Y, T₀, …, T_k−1 be algebraically independent over K. We put q = p^h (h ∈ ℕ) and define

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