*(English)*Zbl 0966.11047

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.

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 ${\mathbb{Z}}_{p}$-extension. Since the application of Jacobi sums to primality testing was too beautiful to omit, I have also included it in this chapter.

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.

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.”

For the reader’s convenience we list the chapter headings:

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 ${\mathbb{Z}}_{p}$-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 ${\mathbb{Z}}_{p}$-extension).

Appendix: Inverse limits, Infinite Galois theory and ramification theory, Class field theory.

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.

##### MSC:

11R18 | Cyclotomic extensions |

11-01 | Textbooks (number theory) |

11-02 | Research monographs (number theory) |

11R23 | Iwasawa theory |