Galois theory of special trinomials (Q1884026)

In the paper the author continues his interest in Galois groups and splitting fields of some polynomials over function field \(k(X)\) of positive characteristic \(p\). Let \(q\) be a power of \(p.\) The first sections of the paper are devoted to the trinomials \(F(Y)=Y^{1+q}+Y+X\), \(\Phi(Y)=Y^{q^2-1}+Y^{q-1}+X\) and \(\widehat{\Phi}(Y)=Y^{q^2}+Y^q+XY\) for which the splitting fields are examined. These polynomials have been considered in the earlier papers of the author [Isr. J. Math. 88, 1--23 (1994; Zbl 0828.14014) and Proc. Am. Math. Soc. 125, 1643--1650 (1997; Zbl 0912.12004)]. The second part of the paper concentrates on the Galois group of the \(n\)th iterate \(\widehat{\Phi}^{[[n]]}(Y)\) of the vectorial \(q\)-polynomial of \(q\)-degree \(m\) defined as \(\widehat{\Phi}(Y)=\sum_{i=0}^m\,a_iY^{q^{m-i}}.\) The paper ends with some generalisation of the results for the trinomial \(F(Y)\), \(\Phi(Y)\) and \(\widehat{\Phi}(Y)\) to the case of these trinomials slightly modified.