There is at present no systematic introduction to the basic cyclotomic
theory. The present book is intended to fill this gap. No connection will be
made here with modular forms, the book is kept essentially purely cyclotomic,
and as elementary as possible, although in a couple of places, we use class
field theory.
Some basic conjectures remain open, notably: Vandiver's conjecture that
h+ is prime to p.
The Iwasawa-Leopoldt conjecture that the p-primary part 0fC- is cyclic
over the group ring, and therefore isomorphic to the group ring modulo
the Stickelberger ideal. For prime level, Leopo.1dt and Iwasawa have shown
that this is a consequence of the Vandiver conjecture. Cf. Chapter Vi, .
Much of the cyclotomic theory extends to totally real number fields, as
theorems or conjecturally. We do not touch on this aspect of the question.
Cf. Coates' survey paper [Co 3], and especially Shintani [Sh].
There seems no doubt at the moment that essential further progress will be
closely linked with the algebraic-geometric considerations, especially via the
Fermat and modular curves.
I am very much indebted to John Coates, Ken Ribet and David Rohrlich
for their careful reading of the manuscript, and for a large number of
suggestions for improvement.