Rotmans Advanced Modern Algebra as first algebra book ,opinions? Ok so I have been trying to learn abstract algebra.I have started with Dummit and Fottes Abstract algebra, but to me it seems to be burdened with informal developments,and proofs which are not outlined as such,thus it makes it complicated to study out of it(subjective opinion of course)
Now I have been looking at Rotmans Advanced Modern algebra as one to try next(I have only reached chapter 4 of Dummits book just to make a note)and it seems really well structured
What is confusing me is fact that there is also a book by Rotman called a First Course in Abstract Algebra,but to me it seems that it really takes it slow,and builds up on quite a bit of number theory.
What are your opinions as to what I should do conisidering that I require the text to be well structures with outlined definitons and potentialy proof?Thanks in advance
 A: Modern algebra is a pretty unique subject. There are a ton of ways to approach the subject, and a lot of different authors do so.
One way that this can be seen is through the separation between symmetry and linear algebra. Some books try to take the focus away from linear algebra, as to not burden students, and put emphasis on symmetry instead.
Another example is the extent in which the author focuses on group theory.
Dummit and Foote's book have quite a lot of group theory in the first half of the book. They even conclude the book with group theory near the end. But it's not entirely needed to do so in a modern algebra course. Rotman only brings up the essentials for group theory and goes on to rings, modules, galois solvability...
Rotman's is a definitely more advanced book than Dummit and Foote's, in my opinion. From a pedagogical point of view, it probably wouldn't be ideal to learn from Rotman first. However, if you are interested in seeing the crisp, clean version without very many examples, it wouldn't be a mistake either.
From my own experience, sometimes when you take the hard route, you can remember the material but see no correlation to what's being studied, why, or how anyone would come up with it. That being said, you could use both books depending on what you wanted to learn! That's always an option too.

For group theory, I would recommend Abstract Algebra by Dummit and Foote, Algebra by Artin (Artin's proofs are sometimes sparse), or the Theory of Finite Groups by Kurzwiel.
For (commutative) ring theory I would recommend Commutative Algebra by Atiyah, and Algebra by Rotman.
As an overall summary of abstract algebra Topics in Algebra by Herstein is faily nice, Algebra by Artin is good (but sparse on proofs at times. Lots of exercises on the other hand), and Algebra by Rotman is certainly more advanced but still good. Rotman also touches on some of the subjects being researched today like representation theory.
All of the above texts have some mention of field theory and galois theory as well.
