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Cyclotomic fields have always occupied a central place in number theory, and the so called "main conjecture" on cyclotomic fields is arguably the deepest and most beautiful theorem known about them. It is also the simplest example of a vast array of subsequent, unproven "main conjectures'' in modern arithmetic geometry involving the arithmetic behaviour of motives over p-adic Lie extensions of number fields. These main conjectures are concerned with what one might loosely call the exact formulae of number theory which conjecturally link the special values of zeta and L-functions to purely arithmetic expressions.
Written by two leading workers in the field, this short and elegant book presents in full detail the simplest proof of the "main conjecture'' for cyclotomic fields. Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. The masterly exposition is intended to be accessible to both graduatestudents and non-experts in Iwasawa theory.
Written by two leading workers in the field, this short and elegant book presents in full detail the simplest proof of the "main conjecture'' for cyclotomic fields. Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. The masterly exposition is intended to be accessible to both graduatestudents and non-experts in Iwasawa theory.
From the reviews: "The author's aim in this book is to present a proof of the so-called Iwasawa Main Conjecture for the pth cyclotomic field ... . The text is written in a clear and attractive style, with enough explanation helping the reader orientate in the midst of technical details. According to the authors, the book is intended for graduate students and the non-expert in Iwasawa theory. I think that also the expert may enjoy reading this kind of unified treatment of such a beautiful theme." (Tauno Metsänkylä, Zentralblatt MATH, Vol. 1100 (2), 2007) "This book was written to present in full detail a complete proof of the so-called 'Main Conjecture' in the arithmetic theory of cyclotomic fields. ... The book is intended for graduate students and the non-expert in Iwasawa theory; however, the expert will find this work a valuable source in the arithmetic theory of cyclotomic fields. The book is very pleasant to read and is written with enough detail ... . The authors have contributed in an important way to Iwasawa theory with this beautiful book." (Gabriel D. Villa-Salvador, Mathematical Reviews, Issue 2007 g)