Students should achieve an advanced mastery of the topics listed on the 586 Math Wiki page. This means that they should know all relevant definitions, correct statements and proofs of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.

1. Number Fields
2. Prime decomposition in rings of integers
3. Ideal Class Group
4. Dirichlet's unit theorem
5. Cebotarev Density Theorem (Statement)
6. Dedekind zeta function