Herstein's Preface to Topics in algebra
The idea to write this book, and more important the desire to do so, is a direct outgrowth of a course I gave in the academic year 1959-1960 at Cornell University. The class taking this course consisted, in large part, of the most gifted sophomores in mathematics at Cornell. It was my desire to experiment by presenting to them material a little beyond that which is usually taught in algebra at the junior-senior level.
I have aimed this book to be, both in content and degree of sophistication, about halfway between two great classics, A Survey of Modern Algebra by Birkhoff and MacLane and Modern Algebra by Van der Waerden.
The last few years have seen marked changes in the instruction given in mathematics at the American universities. This change is most notable at the upper undergraduate and beginning graduate levels. Topics that a few years ago were considered proper subject matter for semi-advanced graduate courses in algebra have filtered down to, and are being taught in, the very first course in abstract algebra. Convinced that this filtration will continue and will become intensified in the next few years, I have put into this book, which is designed to be used as the student's first introduction to algebra, material which hitherto has been considered a little advanced for that stage of the game.
There is always a great danger when treating abstract ideas to introduce them too suddenly and without a sufficient base of examples to render them credible or natural. In order to try to mitigate this, I have tried to motivate the concepts beforehand and to illustrate them in concrete situations. One of the most telling proofs of the worth of an abstract concept is what it, and the results about it, tells us in familiar situations. In almost every chapter an attempt is made to bring out the significance of the general results by applying them to particular problems. For instance, in the chapter on rings, the two-square theorem of Fermat is exhibited as a direct consequence of the theory developed for Euclidean rings.
The subject matter chosen for discussion has been picked not only because it has become standard to present it at this level or because it is important in the whole general development but also with an eye to this "concreteness." For this reason I chose to omit the Jordan-Hölder theorem, which certainly could have easily been included in the results derived about groups. However, to appreciate this result for its own sake requires a great deal of hindsight and to see it used effectively would require too great a digression. True, one could develop the whole theory of dimension of a vector space as one of its corollaries, but, for the first time around, this seems like a much too fancy and unnatural approach to something so basic and down-to-earth. Likewise, there is no mention of tensor products or related constructions. There is so much time and opportunity to become abstract; why rush it at the beginning?
A word about the problems. There are a great number of them. It would be an extraordinary student indeed who could solve them all. Some are present merely to complete proofs in the text material, others to illustrate and to give practice in the results obtained. Many are introduced not so much to be solved as to be tackled. The value of a problem is not so much in coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver. Others are included in anticipation of material to be developed later, the hope and rationale for this being both to lay the groundwork for the subsequent theory and also to make more natural ideas, definitions, and arguments as they are introduced. Several problems appear more than once. Problems, which for some reason or other seem difficult to me, are often starred (sometimes with two stars). However, even here there will be no agreement among mathematicians; many will feel that some unstarred problems should be starred and vice-versa.
Naturally, I am indebted to many people for suggestions, comments and criticisms. To mention just a few of these: Charles Curtis, Marshall Hall, Nathan Jacobson, Arthur Mattuck, and Maxwell Rosenlicht. I owe a great deal to Daniel Gorenstein and Irving Kaplansky for the numerous conversations we have had about the book, its material and its approach. Above all, I thank George Seligman for the many incisive suggestions and remarks that he has made about the presentation both as to its style and to its content. I am also grateful to Francis McNary of the staff of Ginn and Company for his help and cooperation. Finally, I should like to express my thanks to the John Simon Guggenheim Memorial Foundation; this book was in part written with their support while the author was in Rome as a Guggenheim Fellow.
JOC/EFR December 2008
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