Topics In Algebra 2nd Edition

Preface to the Second Edition 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 an 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 book. I should like to take this opportunity to thank them for their help and kindness. v Preface to the First Edition 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 j unior-senior level. I have aimed this book to be, both in content and degree of sophisti­ cation, about halfWay between two great classics, A Survey t.if Modern Algebra, by Birkhoff and MacLane, and Modern Algebra, by Van der Waerden. The last few years have seen marked ch anges 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 m atter 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 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 intro­ duce 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 con­ crete situations. One of the most telling proofs of the worth of an abstract vii Contents 6.5 6.6 6.7 6.8 6.9 6. 10 6. 1 1 7 Canonical Forms: Nilpotent Transformations Canonical Forms: A Decomposition of V: Jordan Form Canonical Forms: Rational Canonical Form Trace and Transpose Determinants Hermitian, Unitary, and Normal Transformations Real Quadratic Forms. 292 298 305 313 322 336 350 Selected Topics 355 7. 1 7.2 7.3 7.4 356 360 368 371 Finite Fields Wedderburn's Theorem on Finite Division Rings A Theorem of Frobenius Integral Quaternions and the Four-Square Theorem xi 1 Preliminary Notions One of the amazing features of twentieth century mathematics has been its recognition of the power of the abstract approach. This has given rise to a large body of new results and problems and has, in fact, led us to open up whole new areas of mathematics whose very existence had 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 into its 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 thought to have been unconnec