abstract algebra - what is the relation between solving a polynomial and the decomposition series of its galois group?

Can someone please explain me what is the exact relation between solving a polynomial by resolvents, and its corresponding galois decomposition series? Where and how do the normal subgroups and the cyclic quotient groups come in and what exact implication does it bring? I have been obsessed with this issue for weeks and I simply cannot grasp it. I am mainly interested in sketching a unified way to solve polynomials of degree 2, 3, 4, and greater than 5 when solvable. Maybe you could give an exact, step by step procedure for solving the cubic, and highlighting what happens at every step (i.e., when I go to a "lower" subgroup from its decomposition series). If you can, a more intuitive approach would be helpful. Not necessarily rigurous.

UPTADE: I am not referring to the theoretical part on field extension, but on the more practical part, how do these concepts come in in actual working out of the solution step by step.

Answer

I've found Dr. Salamone's videos on the subject very helpful. In particular check out his three-part series called A Cubic Workout in which he explains in simple terms how the cubic is solved using its decomposition series.

A Cubic Workout Part 1, Part 2, Part 3.   All three as a list.