Galois Theory is a topic which is, at least in the algebra books I have, covered in the last chapter as the most beautiful result of algebra. I know that Galois introduced group theory and proved that it was impossible to solve an equation of type f(x)=0, where f(x) has a term of x in the 5th degree or higher, by means of a formula. ( Solving the quintic by radicals is how it is described. ) What bothers me is that I still can't follow the proof, or worse: I simply don't get it.
I found a hint though. The Galois Group of x^2-1=0 is C2 and of x^4-2=0 the Galois Group is the Dihedral Group of order 8 ( symmetry group of the square ). Will play a bit with these examples, I hope it will break some ice.
Update: the field we work in is Q.