Take Galois proof of the insolvablity of the quintic. To understand it you first need to understand group theory and then field theory and obviously how these r

Take Galois proof of the insolvablity of the quintic. To understand it you first need to understand group theory and then field theory and obviously how these relate to the rational numbers and the ring of polynomials over the rationals. Next you need to understand field extensions and then you can start to make sense of galois groups. Then you need to understand the concept of solvability for groups and prove it’s equivalence for galois groups to solutions to polynomials. Then finally you need to demonstrate a polynomial of degree five whose corresponding galois group is unsolvable. Now you can hand wave this and get a basic idea of it. But this is also undergraduate mathematics. The abstractions keep building on each other as you go and soon the easy way to explain it is still in terms of other abstractions. This is for instance why research papers in mathematics are impenetrable, because the requisite knowledge takes years to acquire.