Galois Groups of Trinomials

Galois groups of irreducible trinomials Xn + aXs + b [X] are investigated assuming the classification of finite simple groups. We show that under some simple yet general hypotheses bearing on the integers n, s, a and b only very specific groups can occur. For instance, if the two integers nb and as(n − s) are coprime and if s is a prime number, then already the Galois group of f(X) is either the alternating group An or the symmetric group Sn. This significantly extends work of Osada.