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Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot.

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    $\begingroup$ Another good book that does not require much background is Topics in Algebra by Herstein. $\endgroup$ – Sameer Kailasa Jan 12 '15 at 15:47
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    $\begingroup$ amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/… Gallian's algebra is fairly easy to read. $\endgroup$ – Miguelgondu Jan 12 '15 at 15:49
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    $\begingroup$ I was reading Herstein in high school. You need to be more specific about what you mean by "high school math knowledge." And why do you want to learn abstract algebra? Generally, better to start with elementary number theory. $\endgroup$ – Thomas Andrews Jan 12 '15 at 15:50
  • $\begingroup$ One good book you can try (which combines abstract algebra with the suggestion of number theory) is Childs: A Concrete Introduction to Higher Algebra. Another book (aiming higher than childs) you could try (if childs is to easy) is Cox, David A., Little, John, O'Shea, Donal: "Ideals, Varieties, and Algorithms" (this even goes into robotics!) $\endgroup$ – kjetil b halvorsen Jan 12 '15 at 16:14
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    $\begingroup$ I would suggest reading up on linear algebra first (Hefferon's joshua.smcvt.edu/linearalgebra seems to be a good text written in a leisurely pace; my only gripe is its lack of coverage), and then revisiting the texts you formerly found "not OK for self-learning". A lot of abstract algebra is made to generalize parts of linear algebra, and once you know the latter you will find the former a lot more motivated. And you will need linear algebra all the time in abstract algebra. Dummit & Foote seems to be detailed and comprehensive; maybe that's one book you can read then. $\endgroup$ – darij grinberg Jan 12 '15 at 18:05
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Abstract Algebra Theory and Applications Thomas W. Judson

You can also find it online in here.

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    $\begingroup$ I fear this is not ripe for self-study (yet -- there is always a "yet" with open-source projects). Polynomials are defined badly, for example. $\endgroup$ – darij grinberg Jan 12 '15 at 18:00
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Three books I know of that really are high school level are listed below. Although the books thus far listed (Gallian, Herstein, Fraleigh, Pinter, etc.) are fine texts, these are standard upper undergraduate college level textbooks, not books specifically written for good (but not necessarily near genius level) high school students.

Irving Adler, Groups in the New Mathematics. An Elementary Introduction to Mathematical Groups Through Familiar Examples, The John Day Company, 1967, 274 pages.

Francis [Frank] James Budden, The Fascination of Groups, Cambridge University Press, 1972, xviii + 596 pages.

Israel Grossman and Wilhelm Magnus, Groups and Their Graphs, New Mathematical Library #4, Random House, 1964, viii + 195 pages.

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Abstract algebra by John Fraleigh and JA Gallian Contemporary Abstract Algebra

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  • $\begingroup$ Gallian is a pretty nice option for self study, esp. since undergrads often like solutions in the back of the book to check their work (and it has a solution manual iirc). $\endgroup$ – Batman Jan 12 '15 at 17:40
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I'm astonished that this wasn't mentioned:

Learning Modern Algebra From Early Attempts to Prove Fermat’s Last Theorem By Al Cuoco and Joseph J. Rotman

This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily "end up on the blackboard."

The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalizations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the more general work of Galois and Abel on roots of polynomials. Results and proofs are motivated with specific examples whenever possible, so that abstractions emerge from concrete experience. Applications range from the theory of repeating decimals to the use of imaginary quadratic fields to construct problems with rational solutions. While such applications are integrated throughout, each chapter also contains a section giving explicit connections between the content of the chapter and high school teaching.

Table of Contents

  1. Early Number Theory
  2. Induction
  3. Renaissance
  4. Modular Arithmetic
  5. Abstract Algebra
  6. Arithmetic of Polynomials
  7. Quotients, Fields, and Classical Problems
  8. Cyclotomic Integers
  9. Epilog References

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    $\begingroup$ Regarding I'm astonished that this wasn't mentioned, in my case it's too new to be on my radar yet! $\endgroup$ – Dave L. Renfro Jan 12 '15 at 18:08
  • $\begingroup$ @DaveL.Renfro but it's a very good book! $\endgroup$ – user153330 Jan 12 '15 at 18:11
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    $\begingroup$ +1! Just looked at the polynomials chapter and seeing that it does both the correct formal definitions and readable motivations for them. Very good. $\endgroup$ – darij grinberg Jan 12 '15 at 18:20
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I am also in high school and two books I've used and found very accessible are:

1) 'Visual Group Theory' by Nathan Carter (the diagrams and illustrations are excellent, a bit pricey though)

2) 'Book of Abstract Algebra' by Charles C. Pinter (the 'Dover books on Mathematics' series of mathematics books are all worth a look)

I can attach the contents if you want.

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  • $\begingroup$ +1! both of them are accessible to all and are very easy! $\endgroup$ – user153330 Jan 12 '15 at 18:39
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When I was in high school (60 years ago) I stumbled on W. W. Sawyer's A Concrete Approach to Abstract Algebra. Google found it free at https://archive.org/stream/AConcreteApproachToAbstractAlgebra/Sawyer-AConcreteApproachToAbstractAlgebra#page/n5/mode/2up - I've linked to the page that describes why it might be suitable for you.

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    $\begingroup$ I second Sawyer's book and I should have included Sawyer's book in the books I listed. $\endgroup$ – Dave L. Renfro Jan 12 '15 at 18:07
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Try Pinter's A Book of Abstract Algebra. Over half the book is extended problems that you're gently led through as you sort of discover algebra on your own. There are nice applications to computer science, genetics and kinship networks too. You'll have a great time!

Stahl's Introductory Modern Algebra, a historical approach is worth a look for the way it takes you very early on, after a minimum of pain, to a handwaving (but satisfying for me!) understanding of some of the classic results (quadrature of the circle, constructibility of regular polygons). He also covers an offbeat topic -- the cubic equation (i.e. the next thing after the quadratic equation) -- which is surprisingly interesting (even after you apply the formula, it can take some trickery to simplify your result). But I wouldn't embark on Stahl if I wanted to learn the standard results in the standard order; consider it supplementary reading.

Herstein is way harder than either of these. If you're in high school and you can handle Herstein (like, you can work his medium-difficulty problems on your own), you should look for a pro mathematician to mentor you. (At that point you're probably one of the best students your high school teacher has ever seen, maybe he/she could help you find someone.)

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