The equation that couldn't be solved

When I was a graduate student at Edinburgh during a seminar by someone in the maths department. He mentioned that he had an 8th order polynomial to solve, so he said "so by an elementary application of Galois theory I factored the polynomial in two quartic equations." Woow. We were all deeply impressed by this. (Of course I would have just used maple to solve the equations numerically). I have always felt that I should be able to solve equations by Galois theory. I have just read the book "The equation that couldn't be solved" by Livio. This is a popular mathematics book about Galois's theory. There was a lot of generic stuff about symmetry in the book. As usual strings make an unwelcome appearance. Galois died young in a duel, so he is a romantic figure as well. There wasn't enough detail for me to understand why there is no closed solution to polynomials with order higher than 4. I should have read the 79 pages of Galois theory by Emil Artin instead. This is a more hard core book on sums.