I really like the ideas of Galois theory:
that you can think about all the algebraic numbers you can make starting with some set of them
that there is some structure to this set of "algebraically constructed" numbers and you can understand more about the universe by understanding this structure
What I don't really like is :
that the sets are infinite. The groups are infinite and you can never know the exact representation of any non-trivial element.
But I feel that there is so much that could be exposited, rather simply, about these elements, or about related similar elements. It seems that these structures are essentially combinatorial and perhaps a lot of the complexity comes from the composition of permutations, and the redundancy, the ways that they might cancel, by for example two permutations (which might be part of different elements) combining to the identity permutation.
Anyway, I am really interested in this area and have tried a few different books, but I find the ring theory basis kind of too much overhead. I have looked at Fearless Symmetry, but find it good but also a bit too slow and kind of fuzzy. What I really want is to see some clear diagrams about how these permutations act, on real polynomials but also in a more abstract way is okay, but I really want to see some kind of graph that starts to chart the interaction of Galois elements, and exposes some of there structure. I would even hope for some kind of algorithm that constructs, or uses Galois elements. Most of all I would hope there exists or could be created some kind of visualization (a la perhaps Mandelbrot ) ... and if someone would be so kind as to point me in the direction of substantial materials, I would probably try to create such visualization myself.