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I need to fill some gaps in my algebra knowledge. The problem is: While I do realise the importance and utility of the subject, I do not find it appealing. Is there any book around which shows you why the subject is beautiful and stimulate curiosity?

To list some books which I think do that in other fields: Gravitation by Misner, Thorne and Wheeler (not just for the GR part but also for the differential geometry bit), the shape of space by Week for topology, probably visual complex analysis by Needham, differential topology by Guillemin and Pollack.

For some more details: I am trying to learn the amount of algebra usually known by a math graduate student who is not planning to work in abstract algebra. I have been trained as a theoretical physicist so I received fully rigorous courses in linear algebra and finite group theory (taught by mathematicians) but pretty much nothing else (of course I have had a nodding acquaintance with several other concepts).

The need for some more serious grounding in algebra first arose when I was trying to apply some algebraic topology tools. Therefore I am thinking of learning first about modules and rings (leaving aside a more serious study of fields for the moment), then learn a bit of the categorical viewpoint and see the various structures in action by learning a bit of homological/commutative algebra. I was thinking to get the basic on Lang undergraduate algebra, and then move on to the first few chapters of Hilton-Stammbach possibly complemented by Lang grad book and Atiyah-MacDonald.

I will be happy to hear feedback on my study plan as well, but the main point here is if you can find me a book which will make algebra fascinating!

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  • $\begingroup$ Isaacs book of AMS $\endgroup$ – user40276 Sep 21 '14 at 4:35
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I find Jacobson's Basic Algebra quite stimulating.

Volume I is for the undergraduate courses and Volume II is for graduate courses.

I like the prose in these books, written by a master expositor.

I also like Algebra by Mac Lane and Birkoff very much. It will introduce you to categorical ideas in a painless way by the masters.

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I think Godement's Algebra (translation from french) is one of the best introductory books although it isn't well known for english speakers.

Table of contents: 1, 2, 3, 4, 5, 6, 7

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Artin's Algebra is a beautiful book that conveys the reasons why algebra is appealing or interesting. It is not, however, suitable if you are looking to learn the fundamentals of group or ring theory in detail. There are clever proofs, interesting insights, and great exercises.

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Well, on an undergrad level, Herstein's Topics in Algebra worked for me! Rings and a bit of modules in the middle chapters.

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