Proof of Fermats Last Theorem for Given Exponent 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, $n=5,7,13$?
 A: 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
A: 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.
