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¡¡¡¡since the publication of the first edition£¬ 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 u-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.