Good book for a comprehensive course in modern algebra (categories, modules, groups, rings, fields, galois theory, and representation theory)?

Question

I'm searching for a book to finally top off and polish my idea of introductory abstract algebra before I go further. I've done Gallian and some online 'open' courses, but what I want now is a book that builds from first principles and goes all way to beginning graduate topics, but with greater depth. Artin seems to be one popular choice, along with Dummit and Foote. However, I want to get a final recommendation. Thanks in advance!

Edit: My question is different than the one it's being considered to merge with because it is very specific. I'm someone looking to iterate and polish my knowledge of several topics that I have clearly enumerated.

Answer 1

I think $\mathbf{Dummit}$ and $\mathbf{Foote}$, consists a very good textbook for a good introduction with many examples and many good exercises (and also solutions are available online) and goes up to representation theory, algebraic geometry and category theory (the basics of course) which form the contemporary algebraic viewpoint in mathematics.

Another book that you may like as well is $\mathbf{Hungerford's}$, $\mathbf{Algebra}$, which starts from the basics and goes to Galois Theory, Ring and Category Theory too.

Also $\mathbf{Jacobson's}$, $\mathbf{Abstract}$ $\mathbf{Algebra}$ (volume I, II), provide a thorough introduction (and not only in my head) of many aspects in modern algebra and has many things about ring theory (his nickname along his students was Lord of the Rings, if I'm not mistaken :)) and representations in general.

$\mathbf{Rotman's}$, $\mathbf{Modern}$ $\mathbf{Algebra}$ gives as well a thorough introduction to what we call algebra nowadays but it's huge!

Whilst last but not least $\mathbf{Lang's}$ algebra consists a standard textbook to start with and covers many things too.

From all above I would strongly recommend for a starter Dummit and Foote or Lang and afterwards maybe Jacobson's or Hungerford.

I hope that helps! Enjoy!

Answer 2

A standard reference on graduate algebra is Lang's Algebra, though it is accessible to the undergraduate student with enough patience to see trough all the details he often skips. Also, many examples will come from areas outside algebra, such as algebraic topology. If for a specific chapter you're not familiar with the examples, you may find it's exposition a bit dull.

But as a second year undergrad student, overall I'm having a good experience with it, even though I'm using it only for modules and homological algebra. I certainly find it easier to follow than D&F and Artin - Lang is a genius when it comes to writing math on an elegant and compact form.

EDIT: Though I've never read it, many people told me that his other book, Undergraduate Algebra, it's less comprehensive but more introductory in flavor.