TITLE: Why can we prove mathematically that a formula to solve an (n+5) order polynomial does not exist?
QUESTION [44 upvotes]: I understand that the quadratic equation can solve any second order polynomial. Furthermore, equations exist for polynomials up to fourth order. However, without a graduate level degree and a deep understanding of mathematics, is there an explanation of how we can prove that the equations for solutions to a fifth order and above don't exist?
Thanks in advance.

REPLY [3 votes]: At one time I had the idea that if there wasn't one formula perhaps there was a variety of formula that could be selected from; even infinity one of an infinite.
The class I took explained it very carefully.
First it started with a straight edge and compass and computed the domain of calculations that could be done with them; and then showed trisecting the angle was not in that domain.  Then the book went on to construct the domain of radical solutions and showed it didn't include certain roots of fifth order polynomials.
Now there are certainly solutions; but the point is they are not even in the domain of radical algorithms.  No collection of formulas work because none of them can even reach those roots.
And as mentioned above Galois showed clearly which formulas could be solved and which couldn't.
It's like expressing pi as a rational number; you might get close but are never going to get there for a finite rational.