Book suggestions

Book suggestions for undergrad/master students

Suppose you already learnt some abstract algebra (rings, ideals, modules etc.):

1. You should learn Galois theory. I recommend Hungerford's book ``Algebra" Chapter V (try to solve most of the exercises there).

2. More algebraic preparations:
Learn some language of categories, e.g., Hungerford's book ``Algebra" Chapter X.
Learn some commutative algebra. Standard book: Atiyah-Macdonald ``Introduction to Commutative Algebra" (try to solve most of the exercises there).
Learn some homological algebra. For example, Hilton-Stambach: ``A Course in Homological Algebra", Chapter 1-4. (some of Chapter 5, 6 if you like).

3. Venturing into AG and NT.
For AG, can start with Hartshorne ``Algebraic Geometry" Chapter 1-3.
For NT, Local class field theory, Serre: ``Local fields". Global CFT, Neukirch book; and/or Cassels-Frohlich book.
With some AG learnt, can learn Elliptic curves. Silverman, ``The Arithmetic of Elliptic Curves".
Relation between EC and MF, Diamond-Shurman, ``A First Course in Modular Forms".

4. Galois representations and p-adic Hodge theory.
Fontaine--Ouyang's book is a very convenient reference. Also Brinon--Conrad's CMI notes.

5. Other suggestions.
If you are working with me on undergrad/master thesis, you can either find your own thesis topic, or I will most likely suggest some topics directly related to above topics.
I also highly recommend students to actively participate in "代数与数论“暑期学校 and 丘赛。