Iwasawa Theory — Past and Present

Dedicated to the memory of Kenkichi IwasawaLet F be a finite extension of Q.Let p be a prime number.Suppose that F 00 is a Galois extension of F and that r = Gal ( F 00 / F) is isomorphic to Zp, the additive group of p-adic integers.The nontrivial closed subgroups of r are of the form r n = I'Pn for n ?: 0. They form a descending sequence and r /r n is cyclic of order pn.If we let F n = F~n, then we obtain a tower of number fieldssuch that Fn/ F is a cyclic extension of degree pn and F 00 = Un Fn.In 1956, at the summer meeting of the American Mathematical Society in Seattle, Iwasawa gave an invited address entitled A theorem on Abelian groups and its application to algebraic number theory.The application which he discussed is the following now famous theorem.