By Dr. Serge Lang (auth.)
Kummer's paintings on cyclotomic fields prepared the ground for the improvement of algebraic quantity idea normally by means of Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. notwithstanding, the good fortune of this basic concept has tended to vague specific proof proved by means 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, apart from a number of papers, between that are these through Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. within the mid 1950's, the speculation of cyclotomic fields used to be taken up back via 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 ordinarily, Galois extensions of quantity fields whose Galois workforce is isomorphic to the additive crew of p-adic integers. Leopoldt focused on a hard and fast cyclotomic box, and proven quite a few p-adic analogues of the classical advanced analytic category quantity formulation. particularly, this led him to introduce, with Kubota, p-adic analogues of the complicated L-functions hooked up to cyclotomic extensions of the rationals. eventually, within the past due 1960's, Iwasawa [Iw 1 I] . made the elemental 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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Extra resources for Cyclotomic Fields
If I is an ideal of R, we let 1m = In Rm. card G= IGI. s(G) = (1. L aeG For any, E R we have ,s(G) = (deg ')s(G). If J is an ideal of R, we write d = deg J to mean that d is the smallest integer ;::: 0 which generates'the Z-ideal of elements deg , with, in J. 43 2. Stickelberger Ideals and Bernoulli Distributions Let B/c(X) be the kth Bernoulli polynomial. We let ()/c(N) = ()~(N) = N/C-l Ia~ B/c( <~) )a;l - N"-l Ia~ (B/c( <~») B/c(o»)a;l where B" = Bk(O) is the kth Bernoulli number. We have: deg () =1= 0 and deg ()' =1= 0, for k even.
N R) = w. Proof We define a homomorphism 1 T: R8-+- ZjZ w by mapping an element of the group algebra on its first coefficient mod Z. In other words, if we let TIX = a(I). Note that T(8) 1 1 == In - 2 (mod Z), and therefore that T is surjective. It now suffices to prove that its kernel is R8 n R. But we have whence for odd b prime to m, and IX T(UbIX8) E R, we get == bT(1X8) (mod Z). 29 2. () also lies in R, thereby proving the lemma. We now assume that m = pn is a prime power. / = R() () R is called the Stickelberger ideal We want to determine the index Define for any character Xon Z(m)*.
The sum over a, b, c is taken over elements in Z mod d. The term for which a = b = c = 0 yields a contribution of q2, that is the number of points on the line in F. Next, suppose that in the remaining sum, one of a, b, c is 0 but not all are o in ZjdZ. Say a = 0 but b # O. Then we may write the sum L u+v+w=o L certain 'U, W Xa(u)Xc(w) L all veF Xb(V), and the sum on the far right is O. This shows that all the terms in the sum 23 1. Character Sums with one, but not all, of a, b, c equal to 0 give a contribution O.