Book for abstract algebra (group and galois theory)

[Avatrin]

Hi

I recently read a book called "The fundamental theorem of algebra" by Fine and Rosenberger. It focused specifically on polynomials, and proved the theorem using several fields of mathematics; Two of the proofs were algebraic.

Abstract algebra has been very difficult for me; Mostly because it is hard to think in terms of groups (I reckon fields and rings will get easier as well once I get a firm grip on the group). It is a very abstract concept that is hard to visualize. Also, it is very different than anything I have encountered before (like metric spaces or measure theory).

The aforementioned book is structured as a textbook; It has exercises and teaches the theory needed to understand the proofs. I think by focusing on something familiar, this book somehow made it easier for me to get a better grasp of abstract algebra. However, since it focuses on the fundamental theorem of algebra, its scope is limited. It barely goes through some core concepts in abstract algebra, and it skips a lot of theory I need.

I am having trouble properly understanding factor groups, the isomorphism theorems, Sylows theorems and Galois theory. Most books I have encountered barely dedicate a chapter to each of these concepts.

I think the isomorphism theorems and Sylows theorems are the ones I am struggling with the most. Does anybody know of a book on abstract algebra that has an approach that sounds similar to the one I have described above? Maybe it focuses on construction problems? Or something else I should be relatively familiar with?

 

[fian]

I suggest "Contemporary Abstract Algebra" by Joseph A. Gallian. I think it's quite easy to understand this book. I read this book when the first time I learned abstract algebra

 

[Math Amateur]

Two algebra books that have particularly clear and basic approaches to abstract algebra, together with a good number of examples, are the following:

"A First Course in Abstract Algebra: Rings, Groups and Fields" by Marlow Anderson and Todd Feil (Chapman and Hall/CRC, 2005)

and

"Elements of Modern Algebra" by Linda Gilbert and Jimmie Gilbert (Brooks/Cole, 2009)

Hope that helps,

Peter

 

[mathwonk]

groups are very concrete things and usually arise as symmetries, e.g. the collection of all rotations leaving a platonic solid invariant. in my opinion fields and rings are more elementary than groups because they are commutative. e.g. the addition in a vector space is commutative, but the collection of invertible linear transformations of that space is a non commutative group, and much more complicated than the vector space with its addition (and scalar multiplication) operation. one nice book on abstract algebra featuring symmetries and linear transformation groups is Algebra by M. Artin, but it may be a bit dense as a beginning book. by the way i do not think there are any purely algebraic proofs of the FTA. the one in that book using galois theory e.g. assumes that an odd degree real polynomial has a real root, an analytic fact.

 

[Avatrin]

groups are very concrete things and usually arise as symmetries, e.g. the collection of all rotations leaving a platonic solid invariant. in my opinion fields and rings are more elementary than groups because they are commutative. e.g. the addition in a vector space is commutative, but the collection of invertible linear transformations of that space is a non commutative group, and much more complicated than the vector space with its addition (and scalar multiplication) operation. one nice book on abstract algebra featuring symmetries and linear transformation groups is Algebra by M. Artin, but it may be a bit dense as a beginning book. by the way i do not think there are any purely algebraic proofs of the FTA. the one in that book using galois theory e.g. assumes that an odd degree real polynomial has a real root, an analytic fact.

Yes, I am aware of that. Both algebraic proofs assume continuity, another analytic fact.

However, I think all of the current replies kind of miss the point. This is hard to explain, but let's try..:
The problem I am having is not quite getting abstract algebra; When I read a proof, I get why it is true. The problem arises with "processing it" myself and remembering it. So, when I need a result from a previous chapter to understand a new concept, it becomes hard to understand.

This is probably not making much sense. So, let's try another way of explaining it. There has been almost two years since I had complex analysis. So, I have forgotten a lot. However, if I am given a week or so, I can probably deduce and prove some of the central results from that course, like the residue theorem or Picard's theorem. On the other side, there has been only a few days since I read Euler's totient theorem. That day I could explain why the proof I read was correct. However, I am unsure if I can explain that theorem again today. I know what the totient function is; I know that the theorem says, but I cannot prove it although I completely got the proof completely when I read it. The reason I get it, is because I see the algebra, and it is correct (i.e. it follows the rules I know are correct). It just does not click on a more intuitive level like real analysis or partial differential equations do.

The problem lies in the abstraction. The reason I brought up the FTA book was because when everything was phrased in terms of polynomials, it became easier. Then going back and using the more general versions of the same theorems also became relatively easy. However, that book skips a lot of proofs. Also, it does not cover everything I need.