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I am looking for a good undergraduate level book on Abstract Algebra. By a 'good book' I mean a book which gives equal importance to both, rigor and the historical perspective of the subject.

For example, I am currently reading the book Elementary Number Theory by David M. Burton. Being highly influenced by the representation of the book, I am now looking for books in other areas which follow the same style.

What will be the suggestions?

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3 Answers 3

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For what it's worth, my algebra notes at http://www.math.umn.edu/~garrett/m/algebra/ (also available as a physical "published" book) are fairly down-to-earth, and include some things usually neglected in contemporary "algebra" texts, such as Lagrange resolvents (for solving equations in radicals when possible), discussion of some basic number-theoretic motivations and applications, brief intro to the-most-basic set theory, and emphasis on mapping properties (a.k.a., naive category theory) throughout. Many examples.

I wrote those notes up as companion for an intro abstract algebra course I gave, to document the topics actually covered, which were things I thought (after doing this for a few decades) people should actually know. That is, some of the usual laundry-list of iconic topics were left for a later time, the idea being to have some coherence to the whole, and not pass over important basic examples too quickly, or just leave them as exercises.

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I love the algebra notes you have written. The section on Cyclotomic Polynomials have taught me really slick techniques in proving how certain polynomials are irreducible. No other algebra textbook I have seen actually have worked examples in explaining why, for example, the polynomial $x^6+x^5+x^4+x^3+x^2+x+1$ is irreducible over $\mathbb{F}_3[x]$. Bravo! –  Prism Sep 16 at 5:46
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@Prism, thanks for the encouragement! I did attempt to include examples that are often given as "exercises" while lacking any "model" or "prototype", all too often leaving students with partial solutions, or no solutions, or bad solutions. –  paul garrett Sep 16 at 13:48

A historical perspective on algebra means a very heavy focus on Galois theory, starting from early ideas on permuting the roots of polynomials. That's almost exactly backward from how nearly all texts handle Galois theory, and how they order the subject, so you're ruling out most of the standards immediately. (Not that that's necessarily a bad thing. When I took algebra for the first -- and second -- time, the Galois theory section started with "now you're going to learn why there's no quintic formula, and why you can't do certain geometric constructions," but then we just learned a bunch of seemingly unrelated stuff about fields that I didn't actually learn because it seeemed irrelevant.)

I think I've heard of a book called something like "The Abel-Ruffini Theorem in Problems" which went along these lines and was aimed at talented high-school or undergrad students. Haven't read it, but heard it recommended highly. It probably would have been written in Russian initially and translated into English. [UPDATE: It's "Abel's Theorem in Problems in Solutions" by V. B. Alexseev; thanks MJD!] Unfortunately I must have the title a bit wrong because it's not turning up when I Google it.

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I believe you are thinking of Abel's Theorem in Problems and Solutions: Based on the lectures of Professor V.I. Arnold, by V.B. Alekseev. I give this book an enthusiastic thumbs-up, although I can't say anything more concrete, not having looked at it in several years. –  MJD Aug 29 at 14:10

I am not familiar with Burton. However in my opinion, the two best undergrad books are Artin's "Algebra" for groups, and Dummit and Foote for rings and fields (and more).

You may also benefit from these most excellent free video lectures by B.Gross. In them you will find many colorful historical references:

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

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Really? I felt like Dummit and Foote's coverage of groups was okay but that the section on rings was just a march through a whole bunch of definitions that came too quickly to get any idea about them. Aluffi's book is written with algebraic geometry in mind, which makes the sections on basic commutative algebra much easier to understand. –  Daniel McLaury Aug 29 at 13:46

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