Kummer's paintings on cyclotomic fields prepared the ground for the advance of algebraic quantity thought as a rule by way of Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. despite the fact that, the luck of this basic concept has tended to vague designated evidence proved by way of Kummer approximately cyclotomic fields which lie deeper than the overall idea. For a protracted interval within the twentieth century this point of Kummer's paintings turns out to were mostly forgotten, with the exception of a number of papers, between that are these through Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. within the mid 1950's, the idea of cyclotomic fields used to be taken up back by means of Iwasawa and Leopoldt. Iwasawa considered cyclotomic fields as being analogues for quantity fields of the consistent box extensions of algebraic geometry, and wrote an excellent series of papers investigating towers of cyclotomic fields, and extra as a rule, Galois extensions of quantity fields whose Galois crew is isomorphic to the additive crew of p-adic integers. Leopoldt focused on a set cyclotomic box, and proven quite a few p-adic analogues of the classical advanced analytic classification quantity formulation. particularly, this led him to introduce, with Kubota, p-adic analogues of the advanced L-functions hooked up to cyclotomic extensions of the rationals. eventually, within the past due 1960's, Iwasawa [Iw eleven] made the basic discovery that there has been a detailed connection among his paintings on towers of cyclotomic fields and those p-adic L-functions of Leopoldt - Kubota.

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