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Where can I find reasonably short and elementary proofs (using basic concepts of ring, field, galois theory) of Fermat's Last Theorem for specific $n$? For example, $n=5,7,13$?

Javi
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  • Do you consider being a "unique factorization domain" elementary? – RghtHndSd Dec 13 '13 at 22:44
  • Yes very elementary, by elementary I mean what one can find in a basic one year course on these topics. sorry I should have been more explicit –  Dec 13 '13 at 22:45
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    You might start with a look at http://en.wikipedia.org/wiki/Proof_of_Fermat's_Last_Theorem_for_specific_exponents and the references therein. – Barry Cipra Dec 13 '13 at 22:46
  • by elementary I do not mean trivial, I actually want to see ring theory and field theory all over the place! the shorter the better. the proofs on wiki are utterly ugly, I want conceptual proofs, you see –  Dec 13 '13 at 22:48
  • There is also "Fermat's Last Theorem for Amateurs", by Paulo Ribenboim, although I am not sure it has elementary proofs for orders 5, 7, 13. Still a nice introduction to the mathematics behind "classical" research on FLT (roughly up to Kummer) – Jean-Claude Arbaut Dec 13 '13 at 22:59
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    George Lowther has offered a fantastic proof of FLT for $n=5$ using the ring $\mathbb{Z}[\phi]$; see http://math.stackexchange.com/a/18660/785 for the details. – Steven Stadnicki Dec 13 '13 at 22:59
  • is there no way of avoiding infinite descent? –  Dec 14 '13 at 00:03

2 Answers2

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You didn't specify the level you were looking for. I agree with Old John's idea of H.M. Edwards' "Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory", which is a great read. For some nice free online examples, one could turn to this:

Sophie Germain and Special Cases of Fermat's Last Theorem

which is a nice simple read of particular case. It's also nice to see the work of a female mathematician appreciated/highlighted for once as well.

Another good read is the following (albeit more advanced):

Kummer’s Special Case of Fermat’s Last Theorem

though not 'professionally' written. The cases for regular primes can be read here:

Fermat’s Last Theorem for Regular Primes

A nice historical overview with some cases given some attention is given here:

Introduction to Fermat's Last Theorem

Finally, for an overview of the actual proof of Fermat's Last Theorem (and I mean only and overview as the real proof is very long indeed) can be found here:

An Overview of the Proof of Fermat’s Last Theorem

mathematics2x2life
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Not necessarily elementary (possible you are being over-optimistic?), but a decent place to start might be

H. M. Edwards - "Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory"

He talks about the cases $n=5, 7, 14$ in chapter 3, and he avoids methods involving such esoteric topics as Galois cohomology, and gives copious references, which will probably get you to some accessible proofs.

In my copy, he covers the proof for $n=5$ in some detail in pages 65-73, and gives an outine of Dirichlet's proof for $n=14$ in the exercises at the end of chapter 3.

Old John
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    This answer isn't really specific. Where does that book cover proofs for specific values of $n$? – Martin Brandenburg Dec 13 '13 at 22:47
  • was going to say the same thing about over-optimistic, but it is perhaps less well-known than it should be that Ernst Kummer made huge progress with the Fermat problem way back in the 19th century, solving it for a whole class of primes which are now called "regular". a short discussion of his approach is in http://arxiv.org/pdf/1307.3459v1.pdf – David Holden Dec 13 '13 at 22:49
  • Galois cohomology is very down to earth. Also I prefer books to arxiv articles written by some students (no offence). –  Dec 13 '13 at 22:56
  • @MartinBrandenburg Is it now a little more specific? – Old John Dec 13 '13 at 22:58
  • Yes. Alright :-) – Martin Brandenburg Dec 13 '13 at 23:04