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I'm looking for a reference text with classic results in algebra, like

  • Fundamental theorem of finitely generated abelian groups
  • Every field has a unique smallest prime field, which is either $\mathbb{F}_p$ for some prime $p$ or $\mathbb{Q}$.
  • Every two $K$-vector spaces with isomorphic bases [as sets] are isomorphic [as vector spaces] (NOT just the finite case)

I want something to cite for a class project. I have Lang's Algebra, and like it, but it seems some of these things are too basic for him. Artin's Algebra doesn't seem to have the infinite case of the last point (maybe I just didn't look carefully).

Is there anything that's organized in encyclopedic fashion like Wikipedia?

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"Basic Algebra" by Knapp may be the best for your modest needs. –  Mark Dec 7 '11 at 22:19
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@KaratugOzanBircan : I think Mathematics 250A from Ken Ribet is useful, too. Here is the link to it: Maths 250A –  Nguyễn Duy Khánh Dec 8 '11 at 4:07
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@KaratugOzanBircan, thanks, the MacLane book looks interesting. –  gatoatigrado Dec 8 '11 at 4:56

3 Answers 3

up vote 10 down vote accepted

Abstract Algebra from Dummit and Foote is a very nice abstract algebra book which covers a lot but not as general as Lang's Algebra.

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Absolutely agree with Jwaixs, but on the other hand I think Abstract Algebra by Jean Antoine Grillet, GTM 242, Springer is also a good one(it provides something that are not included in the other textbook, like hereditary ring, etc..), especially if you can do all the exercises :D

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I consider Grillet to be the best one-volume text for a graduate algebra class that currently exists. Much shorter,more readable and focused then Lang and very similar in spirit. –  Mathemagician1234 Dec 8 '11 at 2:57
    
I agree, Grillet is my favourite Abstract Algebra book, by far. It isn't perfect, but it's much better, IMO, than Lang, Rotman, Knapp, Aluffi, Hungerford, Dummit&Foote, etc. –  Leon Dec 10 '11 at 17:54
    
@LeonLampret : I think Dummit & Foote is better for beginner than Grillet's one. Both books have its own advantages and in my opinion, they are both best books on Abstract Algebra. –  Nguyễn Duy Khánh Dec 11 '11 at 4:50

The author Prof. Louis Rowen has excellent level books in Abstract Algebra: Group theory, Ring theory , Rings with Polynomial Identities, etc.

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