To show: Galois group of a polynomial is not an alternating group .

Question

I have a polynomial of this type: $p(t)+l(t)s(a,t)$ in $F_q(a)[t]$, where a=(a_0,...,a_m)$ are specialization.

I could show that this polynomial is doubly transitive, by making $a_0=0$ (specialization technique for irreducibility and assumptions on f and g).

Now I need to show that the discriminant of this polynomial is not a square. Here I do not know how to proceed. Could any one please help me