## Elements of Abstract AlgebraThis 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. |

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In my opinion this is a very bad book for an undergraduate or even graduate student, unless the student is very very good at math and also very motivated. It very little expository material and is mostly all problems. A better choice would be the book by Beachy and Blair, or the book by Fraleigh.

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One of the best ways for interested undergraduates to get a solid foundation in modern algebra. The book is essentially a problem book, with minimal expository text and no solutions at all, but they are well-chosen, well-motivated, and very relevant exercises.

Also, the book is small and very cheap, it deserves a place on every math shelf.

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### Common terms and phrases

a e G abelian group algebraic integer automorphisms called Clearly coefficients complex numbers composition series conjugate Consequently constructible contains Corollary cyclic group Dedekind domain defined denote the set divides element of G equation equivalence relation euclidean domain extension of F field F finite extension finite group fixed field follows fractionary ideal Furthermore Galois extension Galois group given greatest common divisor group G group of order homomorphism identity element implies integral domain inverse isomorphic left coset Let f Let G linear mapping mathematics minimal polynomial modulo n-th roots natural numbers nonzero normal subgroup number field number of elements one-to-one correspondence permutations prime ideal principal ideal domain Proof proper prime ideal Proposition Prove rational field rational numbers real numbers roots of unity Show solvable splitting field subfield subgroup of G subset Suppose theorem unique factorization vector space zero

### Popular passages

Page iv - WAS just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension or variation of the following arithmetical formula: 2 + 2 = 4.

### References to this book

Abstract Algebra and Famous Impossibilities Arthur Jones,Sidney A. Morris,Kenneth R. Pearson No preview available - 1994 |