This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat’s Last Theorem, and Iwasawa’s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott’s proof of the vanishing of Iwasawa’s f-invariant.
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The job done by the author will surely be valued highly by everyone, whether student or scholar, working in algebraic number theory. -- T Metsnkyl, Mathematical Reviews, issue 97h
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Extraordinarily Comprensive Review,
By "orca137" (Mansfield, PA USA) - See all my reviews
This review is from: Introduction to Cyclotomic Fields (Graduate Texts in Mathematics) (Hardcover)
This book has pretty much anything you could want about the theory of cyclotomic fields: Fermat's Last Theorem(See how it was done before Wiles!), an introduction to iwasawa theory, and a proof of Thaine's theorem for those of you who are interested in Mihailescu's proof of Catalan's conjecture. One thing though, this book is not really an introduction. If you're not familiar with Algebraic Number Theory and p-adic analysis, you will be completely lost in this book. It definitely doesn't hurt to know at least the major statements of Class Field Theory. Other than that, though, a truly great book!
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Inside This Book
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First Sentence:
We start with a special case of Fermat's Last Theorem, since not only was it the motivation for much work on cyclotomic fields but also it provides a sampling of the various topics we shall discuss later. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
real abelian fields, abelian number fields, cyclotomic function fields, relative class number, units modulo cyclotomic units, unités cyclotomiques, prime cyclotomic fields, unités elliptiques, corps abéliens, class number formula, cyclotomic number fields, distinguished polynomial, ideal class group, modulo roots, prime ramifies, irregular primes, cyclotomic invariants, maximal real subfield, archimedean primes, main conjecture, finite cokernel, residue class degree, finite abelian extension, unramified abelian extension, inertia group Key Phrases - Capitalized Phrases (CAPs): (learn more)
Fermat's Last Theorem, Hilbert's Theorem, Weierstrass Preparation Theorem, Therefore Gal, Sinnott's Proof That New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
We start with a special case of Fermat's Last Theorem, since not only was it the motivation for much work on cyclotomic fields but also it provides a sampling of the various topics we shall discuss later. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
real abelian fields, abelian number fields, cyclotomic function fields, relative class number, units modulo cyclotomic units, unités cyclotomiques, prime cyclotomic fields, unités elliptiques, corps abéliens, class number formula, cyclotomic number fields, distinguished polynomial, ideal class group, modulo roots, prime ramifies, irregular primes, cyclotomic invariants, maximal real subfield, archimedean primes, main conjecture, finite cokernel, residue class degree, finite abelian extension, unramified abelian extension, inertia group Key Phrases - Capitalized Phrases (CAPs): (learn more)
Fermat's Last Theorem, Hilbert's Theorem, Weierstrass Preparation Theorem, Therefore Gal, Sinnott's Proof That New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
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100
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- Cyclotomic Fields (Graduate Texts in Mathematics) by S. Lang on 5 pages
- Algebraic Number Theory (Advanced Studies in Pure Mathematics) by John Coates in Back Matter (1), Back Matter (2), and Back Matter (3)
- Algebraic Number Theory by H. Koch in Back Matter (1), and Back Matter (2)
- A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland in Back Matter (1), and Back Matter (2)
- Class Field Theory (Advanced Book Classics) by Emil Artin in Back Matter (1), and Back Matter (2)
See all 100 books this book cites
- Algorithmic Number Theory: 6th International Symposium, ANTS-VI, Burlington, VT, USA, June 13-18, 2004, Proceedings (Lecture Notes in Computer Science) by Duncan Buell on page 279, page 326, and page 441
- Algorithmic Number Theory: Third International Symposium, ANTS-III, Portland, Orgeon, USA, June 21-25, 1998, Proceedings (Lecture Notes in Computer Science) by Joe P. Buhler on page 453, page 462, and page 481
- Analytic Number Theory (Developments in Mathematics) by Chaohua Jia on page 37, and page 125
- Algebraic K-Theory and Its Applications (Graduate Texts in Mathematics) (v. 147) by Jonathan Rosenberg in Back Matter (1), and Back Matter (2)
- Digital Signal Processing Algorithms: Number Theory, Convolution, Fast Fourier Transforms, and Applications (Computer Science & Engineering) by Hari Krishna Garg on page 120, and page 331
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