I am currently doing an independent study in algebraic number theory using Stewart and Tall's Algebraic Number Theory and Fermat's Last Theorem textbook. I have covered the material for the course (list below). However, I have 2 weeks of course left and I was wondering if I could get suggestions of a topic which is important to see in an introduction course to ANT which is not covered in this course.
List of topics covered:
1) field extensions, minimum polynomial, algebraic numbers, conjugates, discriminants, Gaussian integers, algebraic integers, integral basis.
2) examples: quadratic and cyclotomic fields.
3) norm of an algebraic number.
4) existence of factorisation
5) factorisation in $ \mathbb{Q}(\sqrt{d}) $
6) ideals, $ \mathbb{Z} $-basis, maximal ideals, prime ideals
7) unique factorisation theorem of ideals
8) relationship between factorisation of number and of ideals
9) norm of an ideal
10) ideal classes
11) Minkowski convex body theorem
12) finiteness of class number
13) computations of class number
Looks good to me! I wouldn't complain if anyone told me that that list was all they learned in an algebraic number theory course. But since you have some time, a very important technique that's not yet on your list, if you wish to continue in number theory, is completion. The $p$-adics are interesting in and of themselves, and also make lots of computations much easier (e.g. in the theory of cyclotomic fields -- which might be another nice thing to think about if you have just two weeks time).