I will not ask you about the proof of quadratic reciprocity (or Euler's criterion or the Chinese remainder theorem, which are involved), since there are many proofs, and the one we gave is not among my favorites.
Note that many of the above results are fairly involved, so for the longer ones, I may ask you to either sketch the proof or prove one step of the result. If I ask you to sketch or explain why something is true, the more detailed of an explanation you can give me, the more points you will get (to a point).
For example, if I ask you to explain why Z[i] has unique factorization, simply stating that it is a PID will not get you many points. If you say it is a Euclidean domain, and therefore a PID, you'll get more points. If you briefly explain why it is a Euclidean domain first, you'll get full points. (Of course you could also explain this without using the notion of PID's, a we did in Chapter 6, but it's faster to quote the PID result.)
You should definitely expect that I will ask you to explain the determination of primes of the form x^2+ny^2 for at least one of n=1,2 or 3, as well as the n=5 case. You should also expect that I will ask you about the proof of existence and uniqueness of factorization in to prime ideals, as these ideas were the culmination of the course. Thus you should probably think about what answers you would give in advance, and if you have any questions about what the main points of the proof, or if I like your answers, then you should ask me in office hours beforehand.