Elements of Abstract Algebra (Dover Books on Mathematics) 47.3 Edition by Allan Clark (PDF)

Description

This concise, readable, college-level text treats basic abstract algebra in remarkable depth and detail. An antidote to the usual surveys of structure, the book presents group theory, Galois theory, and classical ideal theory in a framework emphasizing proof of important theorems.Chapter I (Set Theory) covers the basics of sets. Chapter II (Group Theory) is a rigorous introduction to groups. It contains all the results needed for Galois theory as well as the Sylow theorems, the Jordan-Holder theorem, and a complete treatment of the simplicity of alternating groups. Chapter III (Field Theory) reviews linear algebra and introduces fields as a prelude to Galois theory. In addition there is a full discussion of the constructibility of regular polygons. Chapter IV (Galois Theory) gives a thorough treatment of this classical topic, including a detailed presentation of the solvability of equations in radicals that actually includes solutions of equations of degree 3 and 4 ― a feature omitted from all texts of the last 40 years. Chapter V (Ring Theory) contains basic information about rings and unique factorization to set the stage for classical ideal theory. Chapter VI (Classical Ideal Theory) ends with an elementary proof of the Fundamental Theorem of Algebraic Number Theory for the special case of Galois extensions of the rational field, a result which brings together all the major themes of the book.The writing is clear and careful throughout, and includes many historical notes. Mathematical proof is emphasized. The text comprises 198 articles ranging in length from a paragraph to a page or two, pitched at a level that encourages careful reading. Most articles are accompanied by exercises, varying in level from the simple to the difficult.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐And this one, is no exception. Dont let the small size of the book fool you; this is a well written book for the uninitiated in the field of Group theory.

⭐This text takes a wonderful problem solving approach, but the text explains rather little in and of itself, largely requiring you to figure it out yourself by doing the problems. Optimally, one would take a standard text (e.g. Herstein, Dummit and Foote) that develops the theory and the like itself, providing examples of what good proofs look like, revealing tricky properties and theorems, etc., and use that as reading with this as a source of problems.If it had more actual explanation and the like I would give it 5 stars, as it is a seemingly ideal text otherwise. But, the terseness and lack of actual writing on the subject will lead some students astray.

⭐The amazon listing doesn’t let you look inside.The main thing that has to be said about this book is that the proofs are few and far between. I don’t think that that is quite clear from the reviews so far. For instance, you would find that its 59 pages of material on group theory is about one-third of the page count for group theory in the more popular standard texts.I can imagine that Clark’s Algebra would be quite useful—even stunning—as a review text, say for a grad student preparing for an algebra qualifying exam, or for a professor wanting to cultivate his own pre-existing knowledge of algebra. I expect it could be useful as a supplementary text or even as the main text for a graduate course, but probably not as the only course material.I can’t give more than one star for its usefulness as a first or only book of algebra, but I defer to the other reviewers regarding its choice of topics and exercises and regarding the elegance of its structure.

⭐I like these types of books because it’s more of a prove it yourself type of approach. This pedagogy is more effective than the standard approach we’re used to and is more beneficial to any aspiring mathematician. I used this as extra practice for my Farleigh text, we used at the university. I personally prefer this dense introduction to the wordy Farleigh.

⭐As the author it would be immodest to comment on the outstanding quality of my book in its original form, but whatever process was used for converting from print to Kindle, it allows misprints to creep in. For example in the statement of the First Sylow Theorem in section 56, the exponent n on the p has been replaced by “. This makes the statement nonsense. The error is not in the print version.The problem of translation from print to Kindle becomes acute in the final chapter. In the print edition ideals of a ring are always in a fraktur font, but in the Kindle edition, they are given in ordinary letters, not even bold-face. This makes for much more difficult reading. In some equations extra symbols are introduced making complete hash of them. As a final insult the article reference numbers are omitted from the Index and the indentations for subentries removed, making it completely useless.Having textbooks on Kindle is a wonderful thing, but some proof-reading is necessary–especially for mathematics texts–along with a reasonable effort to reproduce the text accurately. It is not enough to run it through OCR and hope for the best. If Kindle is to have a chance in the textbook market–as it should–much greater care needs to be taken with technical books.This is a poor example of conversion to Kindle.BUY THE PRINT EDITION AND SKIP THE KINDLE VERSION.

⭐I have a real fondness for this book. I worked through all the problems in the summer before grad school, which was a long time ago now. After many pleasant hours in sunny parks and cafes, I finished with an understanding and appreciation I had not gotten from the usual undergraduate texts. Recently I bought another paperback copy as a gift for a young relative, hoping she has a similarly positive experience.Since Mr. Clark posts and reads comments on Amazon, I wanted to say “Thank you.”

⭐I am very happy with this book. Allan Clark has an excellent choice of topics and it’s well organized. I am *not* saying this just because he was one of my professors back in our dark ages at Purdue, although that’s how I learned about it.

⭐Excellent Text and reference book. I especially liked the treatment of sets and permutations; it leads directly to encryption.CAB

⭐Aus meiner Sicht staubtrocken durchgezogener Pflichtvorlesungsstoff. Kaum Referenzen zu Nebengebieten wie theoretische Physik. Das Buch ist nicht teuer, aber es ist kein “must have”.Excellent book.The problem based approach is definitely most suitable for my learning of this topic.I just wish there was a solution manual.

⭐Un livre complet, concis sans sacrifice de la rigueur. Texte et exercices en parfaite complémentarité. Un beau livre d’algèbre.A recommander à tous les élèves de taupe et de faculté jusqu’en maîtrise.

⭐

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