Yes, you're right, this is about abstract algebra.
You would need to learn some linear algebra, including enough to understand something about linear dependence of vectors and its relation to determinants. You'd also need to learn about polynomials, multiple roots of polynomials and their relation to derivatives, as well as gcd's and lcm's of polynomials. You'd need to understand something about the algebraic closure of a field (or splitting fields) or at least about the complex numbers and the fundamental theorem of algebra if you were willing to restrict yourself to complex numbers. Then you'd need to read about resultants. (The discriminant of a polynomial f is the resultant of f and its derivative f'.) I think a number of facts about resultants and discriminants would also be hard to understand without some knowledge of the basic theorem on symmetric polynomials.
I haven't been able to locate any undergraduate algebra book in English that discusses resultants outside of the exercises. Resultants are discussed in "Basic Algebra I" by Jacobson and "Algebra" in Lang. It's a topic that beings together a number of areas of undergraduate algebra, so it's not easy in that sense.
Edit: The approach suggested only via field theory would leave out some points of view, but might lead to a working understanding of what discriminants are. In any case, I would think of everything I've listed apart from resultants as being prerequisites for Galois theory.