Graduate students often ask what they should study in preparation for a qualifying exam. The answer is of course that definitions, statements of theorems, examples and sketches of proofs of major theorems are the norm in a qualifying exam. However, this information is often not perceived by the student in the same way as it is intended by the faculty member.
To address this disparity, I have included below a number of sample questions to help you prepare for the algebraic aspect of the number theory qualifying exam. These questions reflect only some of my own prejudices, and are not meant to reflect questions which other examiners might ask. These questions are in no sense intended to reflect a comprehensive review of the material, but should give you a good idea of the type and depth of question which you might be expected to answer. In particular, if you are not comfortable with the vast majority of the material reflected by these questions, you are not ready to take the qual.
In the exam itself, expect to be asked questions like those below, but also some which go beyond the bounds of your knowledge, and possibly outside the bounds of the syllabus. It is by probing the boundaries of your knowledge that we ascertain the depth of your knowledge. You are not expected to know the answer to everything we ask, but you are expected to know a majority.
Caveat: These problems have not really been proofread yet, but I wanted them available ASAP.