Let $f(x)$ be a monic cubic polynomial over a field K. Suppose that the discriminant of $f$ is a square in K and K has the primitive third roots of unity. What can one say about the coefficients of f? For the moment, by the theory of Kummer extensions, I can say that the splitting field of f is the splitting field of a polynomial of the form $x^3-c$ for some c in K. What else?
I am interested especially in the case in which K is a finite field.
I'm afraid very little can be said when $K$ is finite. The reason is that there is a unique cubic extension $L$ of degree three (extensions of finite fields are always Galois and cyclic).
Consequently $L=K(\root3\of c)$ is the splitting field of every irreducible cubic in $K[x]$ and any non-cube $c\in K$. Observe that by the same argument the discriminant of every irreducible cubic in $K[x]$ must be a square.