I'm a third year undergraduate student; since this semester I have to write the dissertion, I'm looking for a textbook to prepare properly.
First off, my notions of algebra consist of two courses (Algebra I and II) that covered the basic aspects of groups, rings and modules; and Algebra III, which covered Galois theory following very faithfully Basic Algebra I's chapter 4 (of Jacobson).
Now, I don't know very well how algebra develops after Galois theory, so I cannot have a precise idea of what this book should contain; anyway it seemed to me that the biggest topics of modern algebra are algebraic number theory and algebraic geometry, and I would restrict to the first one. In particular I would like a book exploiting algebraic structures and commutative algebra; in other words, I prefer a more abstract approach, rather than a numerical one. I don't know if I made very clear what I mean, hope so. Thanks in advance
Best Answer
The book on Commutative Algebra by Atiyah and MacDonald is the standard choice. Very well written, short enough, and you will find almost all the necessary for starting with the standard books in algebraic number theory or algebraic geometry. Do not forget the exercises. They are very important in themselves, not only for learning.
Later on you can learn more commutative algebra while learning these other subjects. More advanced books on Commutative algebra are Matsumura's "Commutative Ring Theory" and Bourbaki's "Commutative Algebra", as well as Grothendieck's EGA (that contains a lot of commutative algebra).