Galois Theory?

Question

I need to write a major essay (20 page paper) about the fundamental theorem in Galois Theory. I can include history and explain things leading upto the theory but I am kinda bummed as to how to structure the essay. The theorem itself is less than a page long if written precisely so i dont know what to write for the other 19 pages.
Any help would be great.

Answer 1

This is just about the fundamental theorem? The most obvious way to take up space is to include a proof (which is by no means trivial) and/or some background to understand the theorem. Explaining the concepts of groups, rings, ideals, fields, irreducible polynomials, field extensions, the automorphism group, etc., with examples, should easily take at least about 5 pages of space.

Galois' life has already produced more than 19 pages of written material (probably closer to 1900, for that matter.) Plus, it's really interesting, and always a good addition to any paper on Galois theory, math, or just about any subject imaginable.

Finally, discuss the applications of the theorem and the theory. I imagine you know about the unsolvability of the quintic by radicals; you can also discuss the application of Galois theory (specifically, Galois representations) to Wiles' proof of the FLT or other well-known results. There's definitely a LOT of material here.

Answer 2

Let the theorem stand in the middle, half the essay giving background and leading up to it, other half giving consequences. Without group theory, would we have any nuclear physics or cosmology? I remember a book in the QA section of UCSB's library called "The Eightfold Path," a phrase out of Buddhism, because Weinberg (I think) used S8 symmetry to organize nuclear particles. This was a precursor of quarks, you know, the later dark ages. Galois, if I recall, wrote down the basis of this theory and then went out and died in a duel. There's human DRAMA in the story if you want to bring that in. A really good paper could be the beginning of a publishable book and a career popularizing math and science. Check out sciambookclub.com.

Answer 3

Oh baby, you got a great deal here. First you write the theory, then you write applications. And there is no dearth of those.

Answer 4

Can do something about the insolvability of the quintic - the most famous application - and its history.