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It's a very soft question. So, Wiles proved Fermat's last theorem, and what knowledge of which area would be required to understand the theorem?

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This feels like it might be a duplicate, but I can't find any. The creation of algebraic number theory was fueled in part by the quest to prove FLT, so there's an obvious area. – anon Jul 13 '12 at 2:33
Two relevant questions from MathOverflow:… and… – Henry T. Horton Jul 13 '12 at 2:37
Do you want to understand the proof (as in the title of the question) or the theorem (as in the body)? Whichever, please edit so they match. – Gerry Myerson Jul 13 '12 at 3:43
up vote 22 down vote accepted

To understand the proof, you need to have a good background in arithmetic and algebraic geometry (including but certainly not limited to the theory of elliptic curves), commutative algebra, algebraic number theory, and modular forms.

If you have an understanding of these things at a graduate level (say a second or third year graduate student who is beginning research on these sorts of topics), then you will be able to understand the strategy of the proof and some aspects in more detail. If you are at the level of having successfully written a thesis in these sorts of areas, you will be able to have a substantial understanding of the proof.

Having a complete understanding is perhaps even more difficult (as Kevin Buzzard points out in a remark in one of the MO threads linked to above) because the proof uses base change results in the theory of automorphic forms (due primarily to Langlands) which are quite technical to prove, and rely on techniques quite different to most of the techniques that Wiles himself uses.

The standard introduction to the argument is the graduate text "Modular forms and Fermat's Last Theorem" (edited by Cornell, Silverman, and Stevens). Another is provided by the long article titled "Fermat's Last Theorem" of Darmon, Diamond, and Taylor. Of course there are many introductions at a more basic level (see the MO threads linked above), but the two references I've given actually contain many details of the proof.

The upshot is that it's probably not realistic to expect to understand much more than the very big picture strategy of the proof unless you are a graduate student focussing on this part of number theory.

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