The Galois groups of the polynomials \(X^ n+aX\ell +b\) (Q1086288)

In this paper criteria for some trinomials of the type \(X^ n+aX^{\ell}+b\) to have Galois group over \({\mathbb{Q}}\) isomorphic to the symmetric group \(S_ n\) are given. The author establishes some known lemmas about generation of \(S_ n\) and of some intertia groups of prime ideals [c.f. \textit{K. Uchida}, Tôhoku Math. J., II. Ser. 22, 220-224 (1970; Zbl 0205.354)], in order to obtain these criteria. A conjecture of Williams-Uchiyama concerning the irreducibility of the polynomial \(X^ n+X+a\) modulo p is proved.