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Galois groups of irreducible trinomials X n + aX s + 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 A n or the symmetric group S n . This significantly extends work of Osada.