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Book Reviews

Review of The Beginnings and Evolution of Algebra

Editorial review
Exceptional complex Lie groups have become increasingly important in various fields of mathematics and physics. As a result, there has been interest in expanding the representation theory of finite groups to include embeddings into the exceptional Lie groups. Cohen, Griess, Lisser, Ryba, Serre and Wales have pioneered this area, classifying the finite simple and quasisimple subgroups that embed in the exceptional complex Lie groups. This work contains the first major results concerning conjugacy classes of embeddings of finite subgroups of an exceptional complex Lie group in which there are large numbers of classes. The approach developed in this work is character theoretic, taking advantage of the classical subgroups of $E_8 (\mathbb C)$. The machinery used is relatively elementary and has been used by the author and others to solve other conjugacy problems. The results presented here are very explicit. Each known conjugacy class is listed by its fusion pattern with an explicit character afforded by an embedding in that class.

Review of Adventures in Arithmetic for the Pre-Algebra Student

Editorial review
Proceedings of the Summer Research Institute on [title] held at the U. of New Hampshire, Durham, July 1988. During the last 20 years operator theory has developed in several directions, using new and powerful methods that have led to the solution of previously inaccessible basic problems. Some of these developments connect with other areas of mathematics, including algebraic topology and index theory, complex analysis in one and several variables, and probability theory. There have been direct applications to systems theory, complex variables, and statistical mechanics. These papers summarize progress and examine the common points of view that now run through the subject. Annotation copyright Book News, Inc. Portland, Or.

Review of Algebra

Editorial review
This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach---emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s---was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors for this printing.

Reviewed by Pedro Lauridsen Ribeiro, from Cotia, SP Brasil
After getting frustated by nearly all the so-called "authoritative" books on abstract algebra (Lang, Hungerford, Jacobson), I really can say that MacLane/Birkhoff is the best die-hard classic on algebra. Now I must stress that this book IS NOT out-of-print: the third edition is actually published by AMS/Chelsea.There's an interesting thing about the evolution of this book: the first edition has become famous among mathematicians, because it brought for the first time an elementary exposition of categories and universal constructions, directly from the horse's mouth (MacLane founded the theory of categories together with S. Eilenberg; Birkhoff was the creator of the theory of lattices), which is used as a basic tool throughout the book; it also contained unusual topics such as multilinear algebra and affine and projective spaces, but no Galois theory. The second edition has gained a chapter on Galois theory, but has lost the part on affine and projective spaces. The third edition is the best! It has recovered the part which was lost in the second edition, and had its exposition considerably polished. While most other books expose abstract algebra as a ugly, prawling monster, MacLane/Birkhoff manage to explain quite esoterical topics (many of them created and/or developed by themselves) in a surprisingly natural and tasty way (compare it with the dry, encyclopaedic style of Hungerford and Lang); although quite big, the book supports several ways of reading and teaching its parts without sacrificing clarity. Another great quality: it is INSPIRING, in the sense that it develops a powerful algebraic intuition, which is, in my opinion, the main obstacle one has to face to learn algebra.

Reviewed by profsd@aol.com, from Virginia Beach VA
This text is a very readable presentation of first year graduate abstract algebra. The material is presented with notations similar to that of Jacobsen in his "Basic Algebra" texts, and is useful as a review text for qualifiers, or for independent study.

Review of Elementary Algebra: A Straightforward Approach

Editorial review
This volume contains many of the papers in the area of algebraic geometry presented at the 1984 Solomon Lefschetz Centennial Conference held in Mexico City. It is the first in a three volume series. The conference focused on this topic along with the areas of algebraic topology and differential equations where Lefschetz made significant contributions. The proceedings begin with two interesting articles: A Page of Mathematical Autobiography, that has been reprinted from an early edition of the Bulletin of the AMS, and "Solomon Lefschetz, a biography" by William Hodge, that is reprinted from the Bulletin of the London Mathematical Society.

Review of Pre-Algebra: A Laboratory Workbook

Editorial review
Comprehensive ... remarkably clear and explicit.

Review of College Algebra Workbook: Technology in the Classroom

Editorial review
From the Preface: "An accurate (though uninspiring) title for this book would have been Applications of the Theory of the Modular Forms $\eta(\tau)$ and $\vartheta(\tau)M$ to the Number-Theoretic functions $p(n)$ and $r_s(n)$ respectively. This is accurate because, except in the first two chapters, we deal exclusively with these two modular forms and these two number-theoretic functions. However, at the heart of these particular applications to the treatment of these specific number-theoretic functions lies the general theory of automorphic functions, a theory of far-reaching significance with important connections to a great many fields of mathematics. Indeed, together with Riemann surface theory, analytic number theory has provided the principal impetus for the development over the last century of the theory of automorphic functions ... I have tried to keep the book self-contained for those readers who have had a good first-year graduate course in analysis; and, in particular, I have assumed readers to be familiar with the Cauchy theory and the Lebesgue theorem of dominated convergence."

Review of Basic Mathematics With Pre-Algebra (Test Yourself)

Reviewed by darisier@aol.com, from FT LAUDERDALE FL

Reviewed by a reader, from india
i had a very nice time with solving these problems

Review of Teach Yourself Algebra

Editorial review
Written by experts in the field, this valuable study guide goes into great detail and serve as a valuable resource.

Reviewed by Kersi N. Billimoria, from San Diego, California
I have the Asian paperback edition of this book. This text was a standard in the 1960s for folks who wanted to bursh up on their High School algebra. The format is pedagogical in the British style from the 1940-50s era. I have always found this to be a faithful companion when I find myself forgetting basic algebraic tasks such as simultaneous equations, factoring, etc. The writing is simple and easy to follow. There are numerous exercises to reinforce the material. There is one drawback. I have found quite a few errors in the solutions within the text AND in some of the answers to exercises. However, if one is familiar with the material these shortcomings can be easily excused. If you are just beginning to learn algebra go with "Practical Algebra " by Selby et al. However if you have take the subject before and just want to brush up basic High School algebra get this volume.

Reviewed by a reader, from Hsv ,al
i would suggest it to everyone it really helps you get your way thruogh Math classes Iteches you the basics of Algebra

Review of Using Algebra

Reviewed by a reader, from Cleveland, Ohio
This book is great. It shows everything from Pre-Algebra to real Algebra. I really recommend it. I have a brand new copy.

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