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This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat s Last Theorem, and Iwasawa s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott s proof of the vanishing of Iwasawa s f-invariant.§