Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0966.11047
Washington, Lawrence C.
Introduction to cyclotomic fields. 2nd ed. (English)
[B] Graduate Texts in Mathematics. 83. New York, NY: Springer. xiv, 487 p. DM 94.00; öS 686.20; sFr 83.00 (1997). ISBN 0-387-94762-0

From the preface to the second edition: ``Since the publication of the first edition (1982; Zbl 0484.12001), several remarkable developments have taken place. The work of Thaine, Kolyvagin, and Rubin has produced fairly elementary proofs of Ribet's converse of Herbrand's theorem and of the Main Conjecture. The original proofs of both of these results used delicate techniques from algebraic geometry and were inaccessible to many readers. Also, Sinnott discovered a beautiful proof of the vanishing of Iwasawa's $\mu$-invariant that is much simpler than the one given in Chapter 7. Finally, Fermat's Last Theorem was proved by Wiles, using work of Frey, Ribet, Serre, Mazur, Langlands-Tunnell, Taylor-Wiles, and others. Although the proof, which is based on modular forms and elliptic curves, is much different from the cyclotomic approaches described in this book, several of the ingredients were inspired by ideas from cyclotomic fields and Iwasawa theory.\par The present edition includes two new chapters covering some of these developments. Chapter 15 treats the work of Thaine, Kolyvagin, and Rubin, culminating in a proof of the Main Conjecture for the $p$th cyclotomic field. Chapter 16 includes Sinnott's proof that $\mu=0$ and his elementary proof of the corresponding result on the $\ell$-part of the class number in a $\bbfZ_p$-extension. Since the application of Jacobi sums to primality testing was too beautiful to omit, I have also included it in this chapter.\par The first 14 chapters have been left essentially unchanged, except for corrections and updates. The proof of Fermat's Last Theorem, which is far beyond the scope of the present book, makes certain results of these chapters obsolete. However, I decided to let them remain, for they are interesting not only from an historical viewpoint but also as applications of various techniques. Moreover, some of the results of Chapter 9 apply to Vandiver's conjecture, one of the major unresolved questions in the field. For aesthetic reasons, it might have been appropriate to put the new Chapter 15 immediately after Chapter 13. However, I opted for the more practical route of placing it after the Kronecker-Weber theorem, thus ensuring that all numbering from the first edition is compatible with the second.\par Other changes from the first edition include updating the bibliography and the addition of a table of class numbers of real cyclotomic fields due to R. J. Schoof.''\par For the reader's convenience we list the chapter headings:\par Ch. 1: Fermat's Last Theorem. Ch. 2: Basic results. Ch. 3: Dirichlet characters. Ch. 4: Dirichlet $L$-series ad class number formulas. Ch. 5: $p$-adic $L$-functions and Bernoulli numbers. Ch. 6: Stickelberger's theorem. Ch. 7: Iwasawa's construction of $p$-adic $L$-functions. Ch. 8: Cyclotomic units. Ch. 9: The second case of Fermat's Last theorem. Ch. 10: Galois groups acting on ideal class groups. Ch. 11: Cyclotomic fields of class number one. Ch. 12: Measures and distributions. Ch. 13: Iwasawa's theory of $\bbfZ_p$-extensions. Ch. 14: The Kronecker-Weber theorem. Ch. 15: The Main Conjecture and annihilation of class groups. Ch. 16: Miscellany (Primality testing using Jacobi sums, Sinnott's proof that $\mu=0$, The non-$p$-part of the class number in a $\bbfZ_p$-extension).\par Appendix: Inverse limits, Infinite Galois theory and ramification theory, Class field theory.\par Tables: Bernoulli numbers, Irregular primes, Relative class numbers, Real class numbers. The new updated edition will surely become a standard reference as its forerunner. It is a superb piece of work of interest for students and researchers in number theory.
[O.Ninnemann (Berlin)]

MSC 2000:: *11R18 Cyclotomic extensions
11-01 Textbooks (number theory)
11-02 Research monographs (number theory)
11R23 Iwasawa theory

Keywords: Main Conjecture for the $p$-th cyclotomic field; Stickelberger's theorem; Iwasawa's construction; $p$-adic $L$-functions; cyclotomic units; Fermat's last theorem; cyclotomic fields; Kronecker-Weber theorem; Sinnott's proof

Citations: Zbl 0484.12001

Cited in: Zbl 1229.11135 Zbl 1230.11134 Zbl 1189.11033 Zbl 1202.11051 Zbl 1107.11046 Zbl 1195.11145 Zbl 1108.11082 Zbl 1084.11060 Zbl 1081.11072 Zbl 1042.11078 Zbl 1017.11052 Zbl 0971.11001 Zbl 1038.11070 Zbl 1038.11071 Zbl 0977.11001 Zbl 0964.11055

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]