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May I respectfully ask what the minimal background needed to read Wiles' proof of Fermat's Last Theorem is?

I'm not an expert on number theory, but out of curiosity I wanted to understand - at a cursory level if possible - the outline of the proof.

Thank you to all responders in advance.

My background: Junior-year undergraduate in Theoretical Physics.

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    $\begingroup$ Wiles proof is extremely long and difficult, and you probably won't find the prerequsites in a text-book. However, if you want to understand the idea of the proof there are several good books e.g., "Modular forms and Fermat's last theorem" By Cornell, Silverman, Stevens. $\endgroup$
    – J.C. Ottem
    Commented Feb 7, 2011 at 7:58
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    $\begingroup$ Galois representations, modular forms, L-functions, elliptic curves,... $\endgroup$
    – David Roberts ♦
    Commented Feb 7, 2011 at 8:07
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    $\begingroup$ Well if Angus Macintyre succeeds in his program (rjlipton.wordpress.com/2011/02/03/…), one will only need to know Peano Arithmetic to prove FLT. But it will be a looong proof. $\endgroup$
    – David Roberts ♦
    Commented Feb 7, 2011 at 8:12
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    $\begingroup$ Replace Witten with Wiles here, and you get the general idea: abstrusegoose.com/272 $\endgroup$
    – Lloyd Smith
    Commented Feb 7, 2011 at 9:05
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    $\begingroup$ How odd. I thought Fermat's Last Theorem was proved by Wiles and Taylor. Poor Taylor, already forgotten... $\endgroup$
    – Kevin O'Bryant
    Commented Feb 9, 2011 at 4:25

6 Answers 6

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This is a very hard proof to do for an undergraduate but there are books available. Tthe book "Invitation to Fermat Wiles" (http://www.amazon.com/Invitation-Mathematics-Fermat-Wiles-Yves-Hellegouarch/dp/0123392519) is an exposition on the proof written for undergraduates for example.

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  • $\begingroup$ Excellent link. Thank you very much! :) $\endgroup$
    – Anonymous
    Commented Feb 7, 2011 at 9:46
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Another book is Notes On Fermat's Last Theorem, by the late Alf van der Poorten. It stops well short of giving Wiles' proof, but still gives you some idea of what you're up against.

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    $\begingroup$ I wasn't aware of 'late'. It saddens me to hear that. $\endgroup$
    – Todd Trimble
    Commented Feb 7, 2011 at 12:01
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    $\begingroup$ Sorry to derail the discussion, but I have just read two tributes to Alf, one by Jeffrey Shallit and one in the Sydney Morning Herald. He was a lot of fun to be around and very entertaining as a speaker. I remember a talk he gave at on transcendental number theory where he recalled the story of the rabbi who was told to explain the essence of Judaism while standing on one foot... and that if he, Alf, had to do the same for transcendental number theory, he'd say "an integer which is bounded above by 1 in absolute value must be zero". Which he said on one foot, and the talk was built on that. $\endgroup$
    – Todd Trimble
    Commented Feb 7, 2011 at 17:22
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This set of notes is covering the background and is comparable to Hellegouarche's book in scope.

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    $\begingroup$ +1 These notes are really nice. Thanks. $\endgroup$
    – Adrian Barquero-Sanchez
    Commented Feb 9, 2011 at 4:49
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    $\begingroup$ can anybody update the link? It's dead at this moment... $\endgroup$
    – No One
    Commented Feb 26, 2019 at 18:32
  • $\begingroup$ These notes have vanished from the net and are not available via the wayback machine. I have a copy, so feel free to email me. $\endgroup$
    – Franz Lemmermeyer
    Commented Jan 28, 2023 at 10:22
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Here is a good set of notes by Nigel Boston. I find them very readable and fairly self contained.

http://www.math.wisc.edu/~boston/869.pdf

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Some of the big ideas and connections (with lots of pertinent references) are presented excellently in Fernando Q. Gouvea's "A Marvelous Proof"

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  • $\begingroup$ It seems that the link fails now. $\endgroup$
    – awllower
    Commented Sep 11, 2014 at 6:17
  • $\begingroup$ Link has been updated. $\endgroup$
    – Aeryk
    Commented Sep 11, 2014 at 13:39
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I know about some good books on the direction: First of all, the book Fermat's Last Theorem by Simon Sin is a pretty good book with the most basic needed materials.
Next, the books 13 lectures on Fermat's last theorem and Fermat's last theorem for amateurs by Ribenboim are pretty well and contain advanced elements.
The last but not the least, the book Fermat's last theorem :a genetic introduction to algebraic number theory is an excellent book by Edwards Harold M which ad hoc adjoins a paper by Kummer, and although it doesn't really solve the problem it provides a well background for it, note that it was published before the theorem was formally proved in 1993.
In general, it is not easy to understand the proof or even to just outline it, while BBC program had produced a video about it to introduce this to the public. Perhaps we can better answer your question provided that you let us know how deeper you want to go in and how much you want to know about the proof exactly. Since you are a junior undergraduate in theoretical physics you must be good in analysis, but what about your algebra? Is it pretty good to go through this?
Anyway, thank you for paying attention.

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