This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed ...
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This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic $K$-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization. (GSM/114)
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I am a retired professor of mathematics from a major university. My field of rsearch was algebra. Previous books by Rotman were very attractive and so I was curious about this 1012 page algebra tome. The reason one writes a book is that at some point one thinks one has organized the material in one's mind successfully and putting the ideas down on paper clarifiys or validates this belief. So Rotman apparantly has reached this point in his thoughts about general algebra. I don't think he has anything to offer us that makes this book a needed addition to the literature. First of all the book is too long. It must have taken Rotman years to write this book and we don't have years to absorb all of it. Due to the continual mathematical "flashbacks" I don't think the book is very useful as a general reference. Here's what I recommend: Jacobson's two volumes Basic Algebra I and II have been reprinted inexpensively by Dover at under US$20 each. Volume I can actually be read and is enough algebra background for anyone entering research. Volume II is an excellent reference and supplies more than enough background for a professioal. However I must credit Rotman with very clear explanations of the material and proofs. I just wish this book was up to the level of Rotman's excellent group theory and homological algebra books.
