By Dr. Serge Lang (auth.)

Kummer's paintings on cyclotomic fields cleared the path for the advance of algebraic quantity concept regularly via Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. besides the fact that, the luck of this common concept has tended to vague specific evidence proved by way of Kummer approximately cyclotomic fields which lie deeper than the final conception. For a protracted interval within the twentieth century this point of Kummer's paintings turns out to were mostly forgotten, aside from a couple of papers, between that are these by means of Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. within the mid 1950's, the idea of cyclotomic fields was once taken up back through Iwasawa and Leopoldt. Iwasawa seen cyclotomic fields as being analogues for quantity fields of the consistent box extensions of algebraic geometry, and wrote an exceptional series of papers investigating towers of cyclotomic fields, and extra in general, Galois extensions of quantity fields whose Galois staff is isomorphic to the additive workforce of p-adic integers. Leopoldt targeting a hard and fast cyclotomic box, and demonstrated quite a few p-adic analogues of the classical advanced analytic type quantity formulation. specifically, this led him to introduce, with Kubota, p-adic analogues of the complicated L-functions hooked up to cyclotomic extensions of the rationals. ultimately, within the overdue 1960's, Iwasawa [Iw 1 I] . made the basic discovery that there has been an in depth connection among his paintings on towers of cyclotomic fields and those p-adic L-functions of Leopoldt-Kubota.

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For convenience we let Lemma 2. (i) Let, E R and suppose that (ii) Let, E Rp and suppose that Proof Write, = ,8 E eO' E Zp[G] = Rp. Then, E Rp. Then, EJ. Jp = ZpJ. L z(b)a b with integral coefficients z(b). Then and therefore Nk-l ~ z(b)B~ « b -kN» ) is p-integral. But an elementary formula for Bernoulli polynomials, obtained directly from the definition, gives for an integer b, Nk-l ~N k- 1 (k) B;(b)k-I - Bk (b) -_ L.. k N ;=0 kiN . Comparing the leading term modulo all the lower order terms, and taking into account that Bl = -!

The second, concerning l-adic integers, follows by continuity. The lemma is also valid for p = 2. We may define E~,c by using Bk(X) - Bk(O) instead of Bk(X) in the definition of Ek,c' The lemma shows that /8' c R. For convenience we let Lemma 2. (i) Let, E R and suppose that (ii) Let, E Rp and suppose that Proof Write, = ,8 E eO' E Zp[G] = Rp. Then, E Rp. Then, EJ. Jp = ZpJ. L z(b)a b with integral coefficients z(b). Then and therefore Nk-l ~ z(b)B~ « b -kN» ) is p-integral. But an elementary formula for Bernoulli polynomials, obtained directly from the definition, gives for an integer b, Nk-l ~N k- 1 (k) B;(b)k-I - Bk (b) -_ L..

47 2. Stickelberger Ideals and Bernoulli Distributions §4. General Comments on Indices Let V be a finite dimensional vector space over the rationals, and let A, B be lattices in V, that is free Z-modules of the same rank as the dimension of V. Let C be a lattice containing both of them. We define the index = (C: B). (A . B) . (C:A) It is an easy exercise to prove that this index is independent of the choice of C, and satisfies the usual multiplicativity property (A: D)(D: B) = (A: B). Furthermore, if E is a lattice contained in both A and B then (A .

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