I am taking general abstract algebra soon and am covering most of Dummit and Foote’s book up to Galois Theory. If one wanted to continue their study of abstract algebra after this, would they read Serge Lang’s Algebra or Jacobson’s Basic Algebra I & II? How do they differ and what are pros and cons of each w.r.t. rigor, depth, and relevance?
I only mention these two books because they are on my shelf. Thanks!
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It depends more on the area you want to pursue than on a specific book. One can go into detail in group theory, ring theory or Galois theory, e.g., for infinite Galois theory with profinite groups, or study algebraic number theory, algebraic geometry, commutative algebra, coding theory, cryptography, Lie algebras, representation theory, etc. All of these areas are based on abstract algebra and for each direction there are many standard books available. Like for learning a new language, for the "dictionary part", to learn and to understand the meaning of new definitions, almost every dictionary will do - in particular the ones by Serge Lang or Nathan Jacobson.
It depends what you want to do.
There is a lot of overlap in the topics covered in Lang, Jacobson, and Dummit & Foote. These are all graduate-level books, and so after working through one of these books you are presumably well-prepared to read actual research papers in algebra, or to go on to some more specialized area of algebra like representation theory, some advanced permutation group stuff, classical groups, etc.
In other words, it is likely that working through one of these books will not do a lot to advance your study of abstract algebra. They are better used as a reference to help you study advanced topics you are interested in.