Reviews for Contemporary abstract algebra

The average rating for Contemporary abstract algebra based on 2 reviews is 2.5 stars.

Review # 1 was written on 2017-09-06 00:00:00 Luis Mondragon Before I delve into an analysis of this excellent textbook, let me highlight first that abstract algebra, contrary to much uniformed opinion, is not a specialistic, esoteric field in pure mathematics characterized by very limited applicability to the physical world. On the contrary, this subject is not just extremely beautiful, but a very important and even foundational discipline in many areas of mathematics and science. In fact, structure and symmetry are a fundamental aspect of physical reality. It might well be argued that science is nothing but the discovery of symmetries, patterns and structures that define reality at its most fundamental level. There is for example a natural connection between particle physics and representation theory of abstract algebra, linking the properties of elementary particles to the structure of Lie Groups and Lie Algebras (in particular, the different quantum states of any given elementary particle corresponds to an irreducible representation of the Poincare' group). Poincare' invariance is the fundamental symmetry in particle physics: a relativistic quantum field theory must have a Poincare'-invariant action (one of the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is represented by the Poincare Group, the symmetry group of special relativity). We should also not forget that, while the spacetime symmetries represented by the the Poincare' group are the first that come to mind and the easiest to conceptualize, the physical world is also defined by many other types of symmetries, not least some internal symmetries such as the "color" symmetry SU(3) (the symmetry corresponding to the continuous interchange of the three quark "colors" responsible for the strong force). Moreover, a gauge U(2) × U(1) theory fully explains the unified electro-weak interactions (the requirement that a theory be invariant under local gauge transformations involving the phase of the wave function ultimately brought to the positing of the laws underpinning the interaction of light and matter - indeed, in quantum mechanics, gauge symmetry can be seen as the basis for electromagnetism and conservation of charge). In more general term, local symmetries play a fundamental role in physics as they form the basis for gauge theories (a gauge theory is essentially a type of field theory "in which the Lagrangian is invariant under some Lie groups of local transformations"). It is also important to highlight that symmetry in general is fundamental in the physical world: as an example charge, parity, and time reversal symmetries (CPT) are fundamental symmetries in the Standard Model, whose transformations are symmetric under the simultaneous operation of charge conjugation (C), parity transformation (P), and time reversal (T). Going beyond the realm of fundamental physics, molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties such as its dipole moment. In the biological sciences, I was told (my personal knowledge of the biological sciences is pretty rudimentary) that symmetry and symmetry breaking play a prominent role in developmental biology, from bilaterians to radially symmetric organisms (see for example ). Moreover, group theory has many applications in cryptography, robotics and computer vision, just as another example of concrete application of abstract algebra concepts. Abstract algebra is therefore extremely important and foundational in many fields, and not just in pure mathematics: it provides a set of powerful technical and conceptual tools that allow abstraction of apparently complex systems in such a way as to allow a rigorous mathematical treatment of some of their fundamental and defining features (symmetries and patterns) through the usage of algebraic structures such as groups, rings, vector spaces, algebras and fields. Adding hours on a clock, for example, is like working in a cyclic group; many manufacturing processes might be shown to be isomorphic to products of permutations of a finite group; and group theory and abstract algebra applied to molecular systems biology support the design of new drugs. In general, abstract algebra (and group theory in particular) provides a framework for constructing analogies or models from abstractions, and for the manipulation of those abstractions to design new systems, make predictions and posit new hypotheses. OK, now that justice has been rendered to this fascinating and beautiful subject, the time has finally come for me to review this informative University textbook, which represents a very good introductory treatment of the fundamental elements of abstract algebra (group theory, rings, fields) at a level which is typically addressed at senior Mathematics undergraduate stage. This book is very accurate and nicely written, with very few typos (all of them minor), a very good choice and variety of relevant exercises at different level of complexity (the majority of them are actually pretty easy), with minimal and only occasional hand-waving; it is also a book rich with examples (a critical feature in a subject that sometimes can sorely test the abstract thinking capabilities of even the keenest reader), and presenting a tightly organized and generally reader-friendly progression of concepts and techniques. I also greatly appreciated the full list of notational conventions at the beginning of the book, and the list of suggested readings at the end of each chapter. On the not-so-positive side, I must said that I would have preferred way more depth and detail in some areas such as symmetry groups, that the proofs of some theorems are a bit too terse (leaving conceptual and computational gaps that require a bit of effort to fill), and that too few examples of practical applications are presented, occasionally giving the book a very "abstract" feeling. I must also point out, though, that this book contains one of the best treatments of factor groups, the Lagrange theorem, the Sylow theorems and the fundamental theorem of finite abelian groups that I have come across so far. In summary, my general experience with this book has been quite a positive one; independently of the quality of this particular book, I must also say that studying abstract algebra has been for me a rewarding intellectual journey (I did study some abstract algebra at university, but it was only at introductory level and manly focused on vector spaces), as it is very remarkable how, by starting just with the basic definition of an extremely simple concept such as that of a group, whole worlds of progressively richer and more complex structures, patterns and relationships are progressively unveiled in a process of enthralling discovery. Overall, this is a 4-star book. Not a perfect book by any means, and possibly too basic in some areas, but a solid good textbook recommended to anybody interested in a treatment of this fascinating subject delivered, with a good pedagogical approach, at senior undergraduate level.

Review # 2 was written on 2009-04-29 00:00:00 Eclipse Klokan This book is OK as far as presenting abstract algebra in the usual way to undergrads. Competent explanations of the basics of groups, rings, and fields. Numerous easy exercises, which is fine, although it might be nice if there were more challenging ones too. There are two problems. The first problem is, I don't believe in the purely abstract approach to teaching a first course in algebra which this book uses. This book doesn't give you any real idea what the heck algebra is good for and doesn't provide any real connections to anything else. The attempts to liven things up with silly quotes and bios give the book a condescending middle school type feel. I would rather the author attract readers' interest by showing them the power of the algebraic structures discussed. Groups and rings are extremely powerful concepts, but to read this book you would think they are just a game where you write down some axioms and see what random statements you can prove. (None of this is unique to this book. It seems students are simply not expected to see algebra put to any use until grad school, if at all.) The second problem is, this book sells for $170. There is no excuse for this. It is simply disgusting. No one should buy this book or require it for a course at such an exorbitant price. It doesn't do your laundry or cook you breakfast. A $12 Dover book would do fine.