Reddit reviews Topics in Algebra, 2nd Edition

No, it's not. Math-major algebra was typically taught from something like Herstein. These days, Dummit and Foote seems more popular.

If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):

Basic Algebra by Jacobson

Algebra by Lang

Algebra by MacLane/Birkhoff

Algebra by Herstein

Algebra by Artin

etc

But there are other books that are "essential" to modern readers:

Chapter 0 by Aluffi

Basic Algebra by Knapp

Algebra by Dummit/Foot

I'm not sure if you mean abstract algebra or linear algebra, but if it's the former, I liked Herstein's Topics in Algebra. There's also Abstract Algebra by Herstein as well, which I think is a cheaper slimmed down version. I used these books for self study and found Herstein's exposition, particularly at the beginning of chapters, very helpful. He isn't as verbose as your typical 7th edition mass market textbook author though.

For linear algebra, I hear Axler's Linear Algebra Done Right is good. I haven't read it, but I read his paper "Down with Determinants" which is, I think, written in the same style and enjoyed the alternative perspective a lot.

Herstein's Topics in Algebra is the book I learned both group and field theory from. It's a very easy read with lots of good examples and problems that help you develop and learn about the topics.

Also, the field of quaternions with integer coefficients is pretty cool. You can use it to prove that every natural number can be written as the sum of four squares, almost for free just by examining the field.

Is Herstein out of fashion these days? Excellent book. http://www.amazon.com/Topics-Algebra-2nd-Edition-Herstein/dp/0471010901

There's no single book that's right for everyone: a suitable book will depend upon (1) your current background, (2) the material you want to study, (3) the level at which you want to study it (e.g., undergraduate- versus graduate-level), and (4) the "flavor" of book you prefer, so to speak. (E.g., do you want lots of worked-out examples? Plenty of exercises? Something which will be useful as a reference book later on?)

That said, here's a preliminary list of titles, many of which inevitably get recommended for requests like yours:

1. Undergraduate Algebra by Serge Lang
   
   
2. Topics in Algebra, 2nd edition, by I. N. Herstein
   
   
3. Algebra, 2nd edition, by Michael Artin
   
   
4. Algebra: Chapter 0 by Paolo Aluffi
   
   
5. Abstract Algebra, 3rd edition, by David S. Dummit and Richard M. Foote
   
   
6. Basic Algebra I and its sequel Basic Algebra II, both by Nathan Jacobson
   
   
7. Algebra by Thomas Hungerford
   
   
8. Algebra by Serge Lang
   
   Good luck finding something useful!

For an introductory text, I recommend Herstein's Topics in Algebra. It slowly walks you through groups, rings, vector spaces, modules, fields, linear transformations and other selected topics.

Has plenty of exercises and doesn't skip over any details.

If you are a newcomer to abstract algebra, you might consider using a text other than Dummit and Foote. I used baby Herstein (as opposed to big Herstein) in an undergraduate class and found it to be a good introduction.

Fraleigh is a little bit easier to wrap your head around. Get an old edition (or find it at the library), obviously.

Also, I highly recommend Herstein's Topics in Algebra. Again, try to get it from a university library.

Since your current knowledge is limited to calculus only, your goal seems kind of out of reach, at least in my opinion (but it depends on your progress/motivation). Writing good proofs is not something that you learn in a day by reading notes, it's something that comes with lots of experience reading and writing mathematics.

That being said, if you put a lot of focus on your studies it is certainly possible to learn the basics of algebra pretty fast. Linear algebra is an excellent tool, but it isn't required for learning abstract algebra. You can take both linear algebra and group theory classes at once and see where you want to go from there. It is a beautiful field of study for sure!

I'd strongly recommend Herstein's Topics in Algebra for a very solid introduction to most everything algebra-related. It covers Group Theory, Ring Theory, Vector Spaces and Modules, Fields, Linear Transformations, and some selected special topics.

Herstein!! http://www.amazon.com/Topics-Algebra-I-N-Herstein/dp/0471010901?tag=vglnk-c905-20