Galois groups of a certain class of polynomials I have been working on a problem in Galois Theory:
We consider the polynomial f(X)=a+bX+X^n \in Q[X]. For certain coefficients a and b (in fact, an infinite number) we have $f$ irreducible and the discriminant a square. This implies that the Galois group $G$ of f is a nontrivial subgroup of A_n. (In fact, the cardinal of G is divisible by n.) Can we show, at least in certain cases, that $G=A_n$?

It's true e.g. for $f(X) = X^3 - a X - a$ where $a = 4 k^2+6 k+9$ for integers $k$.  Is that enough "certain cases"?
