GALOIS GROUP OF Xpn+aX+a

Abstract

Let p be an odd prime number, n≥3 be a rational integer, and let f(X)=Xpn+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 Spn, or AGL(1,pn)≤G≤AGL(n,p). We also show that G≃Spn, except possibly when |pnpn−1+ap(pn−1)pn−1| is a square, and for each prime divisor ℓ of a∕p, p divides the ℓ-adic valuation vℓ(a) of the integer a.