INTRODUCTION TO CYCLOTOMIC FIELDS(Graduate Texts in Mathematics, 83)
INTRODUCTION TO CYCLOTOMIC FIELDS(Graduate Texts in Mathematics, 83)
Taylor, M. J.
1983-11-01 00:00:00
61 2 BOOK REVIEWS posets. Shelah's monograph presents the theory of proper forcing more or less as it was worked out. As I said in my introduction, virtually no concession is made to the reader. Though familiar with large parts of the book before it appeared, I have found it hard going whenever I have delved into it. (I have not 'read' it in the sense of starting at page 1 and going through to page 491. It is not that type of book. It is however, a mine of information for reference usage.) For the expert or would-be expert in the area, the book is a must. For anyone curious to see how a first-class mathematician actually produces his stuff, the book should provide some interest, since, as I have said, Shelah has published the material exactly as he first noted it down. But anyone wanting to look into the book should be prepared for a hard time (though undoubtedly a rewarding one). KEITH DEVLIN INTRODUCTIO N TO CYCLOTOMIC FIELDS (Graduate Texts in Mathematics, 83) By LAWRENCE C. WASHINGTON: pp. 389; DM.96.—; US$42.70. (Springer-Verlag, Berlin, 1982.) In recent years Iwasawa's work on the theory of cyclotomic fields has been taken up again and has undergone a considerable expansion producing a number of exciting results. (The two most notable of these are the Ferraro-Washington n = 0 theorem and the recent Mazur-Wiles result proving Iwasawa's main conjecture for cyclotomic number fields.) This book is an account of this work. The author motivates the novice by beginning with Fermat's Last Theorem and then continues to keep him at his ease by means of a rather well-written account of a number of basic facts in algebraic number theory and the theory of L-functions. He is then gradually weaned onto Iwasawa theory by means of p-adic L-functions. There then follow chapters on cyclotomic units, ideal classgroups as Galois modules, Z -extensions and a somewhat singular chapter on cyclotomic fields with class number one. Here the reader is caught by surprise by suddenly having to put down all his newly acquired tools in favour of analytic estimation techniques. In my opinion this really is a very beautiful book. Its various parts have been put together in a careful, and well thought out manner. The author takes great pains to motivate the reader and explain conceptually what is going on underneath the sometimes rather technical calculations and manipulations. Above all else, I admire the way that no loose ends are left: when a new concept is introduced it is illustrated by examples, and, for the most part, when a result is stated the conditions are examined and counter examples are given to show why they cannot be slackened. In particular I would single out the treatment of Leopoldt's Spiegelungssatz for its clarity and elegance. However, while being specific, I should mention that proof of fi = 0 is liable to be a trifle disconcerting to the beginner, being over ten pages of technical calculations; but that is the price you have to pay for such a lovely result! The author provides a good number of exercises, where the reader can test his comprehension and develop ideas further. At the end of the book he gives a very useful and thorough bibliography. I should also point out that, in order that the book be accessible to non-number theorists, there is an appendix dealing with results taken from the classfield theory. In consequence this book can be read and appreciated by a wide audience. M. J. TAYLOR
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngBulletin of the London Mathematical SocietyWileyhttp://www.deepdyve.com/lp/wiley/introduction-to-cyclotomic-fields-graduate-texts-in-mathematics-83-FSLakZKjFT
INTRODUCTION TO CYCLOTOMIC FIELDS(Graduate Texts in Mathematics, 83)
61 2 BOOK REVIEWS posets. Shelah's monograph presents the theory of proper forcing more or less as it was worked out. As I said in my introduction, virtually no concession is made to the reader. Though familiar with large parts of the book before it appeared, I have found it hard going whenever I have delved into it. (I have not 'read' it in the sense of starting at page 1 and going through to page 491. It is not that type of book. It is however, a mine of information for reference usage.) For the expert or would-be expert in the area, the book is a must. For anyone curious to see how a first-class mathematician actually produces his stuff, the book should provide some interest, since, as I have said, Shelah has published the material exactly as he first noted it down. But anyone wanting to look into the book should be prepared for a hard time (though undoubtedly a rewarding one). KEITH DEVLIN INTRODUCTIO N TO CYCLOTOMIC FIELDS (Graduate Texts in Mathematics, 83) By LAWRENCE C. WASHINGTON: pp. 389; DM.96.—; US$42.70. (Springer-Verlag, Berlin, 1982.) In recent years Iwasawa's work on the theory of cyclotomic fields has been taken up again and has undergone a considerable expansion producing a number of exciting results. (The two most notable of these are the Ferraro-Washington n = 0 theorem and the recent Mazur-Wiles result proving Iwasawa's main conjecture for cyclotomic number fields.) This book is an account of this work. The author motivates the novice by beginning with Fermat's Last Theorem and then continues to keep him at his ease by means of a rather well-written account of a number of basic facts in algebraic number theory and the theory of L-functions. He is then gradually weaned onto Iwasawa theory by means of p-adic L-functions. There then follow chapters on cyclotomic units, ideal classgroups as Galois modules, Z -extensions and a somewhat singular chapter on cyclotomic fields with class number one. Here the reader is caught by surprise by suddenly having to put down all his newly acquired tools in favour of analytic estimation techniques. In my opinion this really is a very beautiful book. Its various parts have been put together in a careful, and well thought out manner. The author takes great pains to motivate the reader and explain conceptually what is going on underneath the sometimes rather technical calculations and manipulations. Above all else, I admire the way that no loose ends are left: when a new concept is introduced it is illustrated by examples, and, for the most part, when a result is stated the conditions are examined and counter examples are given to show why they cannot be slackened. In particular I would single out the treatment of Leopoldt's Spiegelungssatz for its clarity and elegance. However, while being specific, I should mention that proof of fi = 0 is liable to be a trifle disconcerting to the beginner, being over ten pages of technical calculations; but that is the price you have to pay for such a lovely result! The author provides a good number of exercises, where the reader can test his comprehension and develop ideas further. At the end of the book he gives a very useful and thorough bibliography. I should also point out that, in order that the book be accessible to non-number theorists, there is an appendix dealing with results taken from the classfield theory. In consequence this book can be read and appreciated by a wide audience. M. J. TAYLOR
Journal
Bulletin of the London Mathematical Society
– Wiley
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