TITLE: Proof of Fermats Last Theorem for Given Exponent
QUESTION [6 upvotes]: 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, $n=5,7,13$?

REPLY [8 votes]: 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