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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.

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I honestly do not understand what you want, really. (And what does à la Manderbrot mean?...) – Mariano Suárez-Alvarez Nov 20 '11 at 16:15
Well, Benoit was into visualizing things. Perhaps there was something out there that showed the Galois connections in a visual manner....a la... – Cris Stringfellow Nov 20 '11 at 17:22
@gotafex: This comes a bit late, but I think you're trying to run before you can walk if you're dealing with infinite Galois groups before having some intuition for the finite Galois groups! Consider restricting your attention to classical Galois theory, i.e. the study of finite extensions of fields. All the groups that crop up in that setting are finite. – Zhen Lin Jan 2 '12 at 5:29
Great. That sounds easier. Field extensions which are finite. I will take a look. – Cris Stringfellow Jan 3 '12 at 18:59
up vote 3 down vote accepted

Take a look at Visual Group Theory by Nathan Carter. It cleverly uses various types of visualizations to develop intuitive understanding of the major group theory concepts (e.g., group structure, generators, cosets, quotients, products, subgroups, homomorphisms, Sylow theorems, etc.) and culminates with a chapter on Galois theory.

For those who find standard textbooks dry and/or difficult, this book is a great way to develop intuition and understanding of some relatively abstract topics. The pace is leisurely, with lots of clear and detailed explanations. Solutions to some of the exercises are included in an appendix. It's also a lot of fun to read.

In addition, Carter has created a free program called Group Explorer that you can use as an aid in developing intuition about groups.

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Great. That is a very good response. Thank you, it sounds perfect and what I was looking for. I appreciate the answer after all this time. I will look it up. Very appreciated. – Cris Stringfellow Dec 13 '12 at 19:24
Glad to be of help. Good luck with your studies! – eipi10 Dec 14 '12 at 7:57
I got this book for Christmas. It is good. Thanks. – Cris Stringfellow Dec 27 '12 at 14:11

I'm not sure exactly what you want, but here are some books on Galois theory that emphasize visual approaches:

Lectures on the Icosahedron, by Felix Klein. Describes the insovlability of the quintic using geometric pictures involving the symmetry group of the icosahedron. Somewhat old fashioned.

The geometry of the quintic, by Jerry Shurman. A modern book on solving quintics, with both a lot of rigorous material and a ton of pictures. I haven't read it, but skimming it make me think it's what you are looking for. It is available for free at Shurman's home page.

Galois' dream, by Michio Kuga. This one is one differential Galois theory, which is a different but related topic, but it might appeal to your geometric desires.

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Thanks David, I think that is as good as I am going to get. I suppose what I want really is X is an element of G, X has structure that is made up of X_1, X_2, X_3, and X_10 which are all elements of G composed of the following structures...Kind of like a graph, with elements of the Galois group as vertices and different types of relations as edges. This would really help me understand what is going on. – Cris Stringfellow Nov 20 '11 at 17:24

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