LAUSR

. Let p be an odd prime number, n ≥ 3 be a rational integer, and let f ( X ) = X p n + aX + a ∈ Z [ X ] be an Eisenstein trinomial with respect to p . We prove that the Galois group G of f ( X ) over the field Q of rational numbers, is either the full symmetric group S p n , or AGL (1 ,p n ) ≤ G ≤ AGL ( n,p ). We also show that G ≃ S p n , except possibly when | p np n − 1 + ap ( p n − 1) p n − 1 | is a square, and for each prime divisor ℓ of a/p , p divides the ℓ -adic valuation v ℓ ( a ) of the integer a .