Let $f\in \mathbb Z[X]$ be an irreducible polynomial. Suppose, the discriminant of $f$ is a perfect square.
Can the galois group of $f$ over $\mathbb Q$ be $S_d$, where $d$ denotes the degree of $f$ ?
Additional question : What can we conclude if the discrimiant is the negative of a perfect square, lets say $-81$ ?
it is suggested to try to prove that the galois group is $A_n$, is the discriminant happens to be a perfect square. It is not mentioned directly, but I assume this is because it cannot be $S_n$ in this case.