Good Algebra Textbook: Basic to Graduate Level

[Cexy]

Can anyone recommend me a good algebra textbook that starts out quite basic and goes up to graduate level? I'm familiar with the following topics:

Elementary group theory e.g. normal subgroups and quotient groups, isomorphism theorems, group actions. Elementary ring theory, e.g. ideals, polynomial rings. Basic representation theory, e.g. characters. Differential geometry, e.g. basic properties of manifolds, Lie groups, curvature and connections, elementary properties of Lie algebras.

I'm looking to learn about:

More advanced group theory, e.g. Sylow theorems, simple groups. More advanced commutative algebra and theory of ideals, perhaps Noetherian rings? More about modules as a generalization of vector spaces. Group rings and connection to representation theory. Galois theory, number fields, more representation theory.

If any of it can be tied into equivariant dynamical systems (i.e. symmetric dynamics) then that would be great as that's what I'm doing my PhD in! Thanks a lot. :)

 

[morphism]

Maybe Dummit and Foote?

Table of contents and preface can be found here.

 

[JasonRox]

I would suggest reading like the first half of a textbook by Gallian or the basic one by Herstein and then start from the beginning from Rotman's Group Theory (Graduate Series from Springer).

If you know the basics well, you heard of normal subgroups and such. Then go right into Rotman's. Surely, that is only group theory.

Then for Ring Theory, I would do something similiar. Read half of an elementary textbook and jump into a full blown graduate textbook (they all start from the beginning anyways).

Also, you don't need to read all of Rotman's textbook. The first half will give you more then you generally need. It's an easy read such that you don't need to solve any problems to keep going. Although, if you don't solve problems, you miss out a lot on comprehending the stuff. The bright side is that if you can't solve the majority of the problems of one chapter, you can still keep going as long as you understood what you read. We all have a our "weak" chapters so it's completely reasonable to assume we will get stuck on a chapter at some point.

 

[quantumdude]

For my undergrad algebra course we used A First Course in Abstract Algebra by Fraleigh, and I liked it a lot. For my grad course we are using Algebra by Steinberger (who is also the prof). I find it very readable, and it's free online.

http://math.albany.edu/~mark/classes/520A/

One grad level book that I would avoid is Hungerford. He almost completely avoids semidirect products, which are extremely useful in classifying groups of a given order.

 

[Cexy]

http://math.albany.edu/~mark/classes/520A/

Thanks, that's exactly what I'm looking for.

 

[mathwonk]

well if that's what you want, i don't suppose you want my notes on algebra, since they cover most of that material in only 100 pages. but i offer them anyway. see my webpage for math 8000 notes.http://www.math.uga.edu/~roy/