Algebra Multilineal y K-Teor´ıa Algebraica Clásica. Decimos que un vector v ∈ V es una Álgebra homológica y álgebra conmutativa. Pages·· I approached rev1smg Topics in Algebra with a certain amount of trepidation. On the whole, I was satisfied with the first edition and did not want to tamper with it. subject of abstract algebra and no student should go through such a course without a good Until recently most abstract algebra texts included few if any applica- tions. However  Herstein, I. N. Abstract Algebra. 3rd ed.
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Reproduction or translation of any part of this work beyond that permitted by Sections or of the United States. Copyright Act abtracta the permission of the copyright owner is unlawful. Library of Congress Catalog Card Number: I approached rev1smg Topics in Algebra with a certain amount of trepidation. On the whole, I was satisfied with the first edition and did not want to tamper with it.
However, there were certain changes I felt should be made, changes which would not affect the general style or content, but which would make the book a little more complete. I hope that I have achieved this objective in the present version.
When the first edition was written it was fairly uncommon for a student learning abstract algebra to have had any previous exposure to linear algebra.
Nowadays quite the opposite is true; many students, perhaps even a majority, have learned something about 2 x 2 matrices at this stage. Thus I felt free here to draw on 2 x 2 matrices for examples and problems.
These parts, which depend on some knowledge of linear algebra, are indicated with a. In the chapter on groups Abstractw have largely expanded one section, that on Sylow’s theorem, and added two others, one on direct products and one on the structure of finite abelian groups.
Sylow subgroup was shown. This was done following the proof of Wielandt. The conjugacy of the Sylow subgroups and their number were developed in a series of exercises, but not in the hersteni proper.
Now all the parts of Sylow’s theorem are done in the text materi9-l. In addition to the proof herwtein given for the existence, two other proofs of existence are carried out. One could accuse me of overkill at this point, algrbra rightfully so. The fact of the matter is that Sylow’s theorem is important, that each proof illustrates a different aspect of group theory and, above all, that I love Sylow’s theorem. The proof of the con-jugacy and number of Sylow subgroups algebrs double cosets.
A by-product of this development is that a means is given for finding Sylow subgroups in a large set of symmetric groups. For some mysterious reason known only to myself, I had omitted direct products in the first edition.
Why is beyond me. The material is easy, straightforward, and important. This lacuna is now filled in the section treating direct products. With this in hand, I go on in the next section to prove the decomposition of a finite abelian abstracra as a direct product of cyclic groups and also prove the uniqueness of the invariants associated with this decomposition. In point of fact, this decomposition was already in the first edition, at the end abstracfa the chapter on vector spaces, as a consequence of the structure of finitely generated modules over Euclidean rings.
However, the case of a finite group is of great importance by itself; the section on finite abelian groups underlines this importance.
Its presence in the chapter on groups, an early chapter, makes it more likely that it will be taught. One other entire section hefstein been added at the end of the chapter on field theory. I felt that the student should see an explicit polynomial over an explicit field whose Galois group was the symmetric group of degree 5, hence one whose roots could not be expressed by radicals.
In order to do so, a theorem is first proved which gives a criterion that an irreducible poly- nomial of degree p, p a prime, over the rational field have SP as its Galois group. As an application of this criterion, an irreducible polynomial of degree 5 is given, over the rational field, whose Galois group is the symmetric group of degree 5. There are several other additions. More than new problems are to be found here.
They are of varying degrees of difficulty. Many are routine and computational, many are very djfficult. Furthermore, some interpolatory remarks are made about problems that have given readers a great deal of difficulty.
Some paragraphs have been inserted, others rewritten, at places where the writing had previously been obscure or too terse. Above I have described what I have added.
What gave me greater difficulty about the revision was, perhaps, that which I have not added. I debated for a long time with myself whether or not to add a chapter on category theory and some elementary functors, whether or not to enlarge the material on modules substantially. After a great deal of thought and soulsearching, I decided not to do so.
The book, as stands, has a certain concrete- ness about it with which this new material would not blend. It could be made to blend, but this would require a complete reworking of the material. Preface to the Second Edition v of the book and a complete change in its philosophy-something I did not want to do.
A mere addition of this new material, as abetracta adjunct with no applications and no discernible goals, would have violated my guiding principle that all matters discussed should lead to some clearly defined objectives, to some highlight, to some exciting theorems. Thus I decided to omit the additional topics.
Many people wrote me about the first edition pointing out typographical mistakes or making suggestions on how to improve the heratein. I should like to take this opportunity to thank them for their help and kindness. 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 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 Algebra, by Birkhoff and MacLane, herstsin Modern Algebra, by Van der Waerden. The last few years have seen marked changes in the instruction given in mathematics hersteinn the American universities.
This abstrata is most notable at the upper undergraduate and beginning graduate levels. Topics that a few years ago were considered proper subject matter for semiadvanced 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 heratein hitherto has been considered a little advanced for that stage of the game.
algebta 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 vii abstrwcta Preface to the First Edition concept is what it, and the results about it, tells us in familiar situations. In xbstracta 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-Holder 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 algebda no mention of tensor products or related constructions. There is so much time hrstein 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 that 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: 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 ahstracta. Finally, Algegra should like to express my thanks to theJohn Simon Guggenheim Memorial Foundation; this book was in part written with their support while the author was in Rome as a Guggenheim Fellow. Nilpotent Transformations 6. A Decomposition of V: Rational Canonical Form 6. One of the amazing features of twentieth abstratca mathematics has been its recognition of the power of the abstract approach.
This has given rise to a large body of gerstein results and problems and has, in fact, led us to open up whole new areas of mathematics whose very existence absrracta not even been suspected. In the wake of these developments has come not only a new mathematics but a fresh outlook, and along with this, simple new proofs of difficult classical results.
The isolation of a problem inl’o herstdin basic essentials has often revealed for us the proper setting, in the whole scheme of things, of results considered to have been special and apart and has shown us interrelations between areas previously abstracts to have been unconnected.
The algebra which has evolved as an outgrowth of all this is not only a subject with an independent life and vigor-it is one of the important current research areas in mathematics-but it also serves as the unifying thread which interlaces almost all of mathematics- geometry, number theory, analysis, topology, and even applied mathematics.
This book is intended as an introduction to that part of mathematics that today goes by the name of abstract algebra. The term “abstract” is a highly subjective one; what is abstract to one person is very often concrete and down-to-earth to another, and vice versa.
In relation to the current research activity in algebra, it could be described as “not too abstract”; from the point of view of someone schooled in the. Be that as it may, we shall concern ourselves with the introduction and development of some of the important algebraic systems-groups, rings, vector spaces, fields.
An algebraic system can be described as a set of objects together with some operations for combining them. Prior to studying sets restricted in any way whatever-for instance, with operations-it will be necessary to consider sets in general and some notions about them.
At the other end of the spectrum, we shall need some informa- tion about the particular set, the set of integers.