By Dr. Serge Lang (auth.)

ISBN-10: 1461299454

ISBN-13: 9781461299455

ISBN-10: 1461299470

ISBN-13: 9781461299479

Kummer's paintings on cyclotomic fields cleared the path for the advance of algebraic quantity thought quite often via Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. besides the fact that, the luck of this basic conception has tended to vague designated evidence proved by means of Kummer approximately cyclotomic fields which lie deeper than the overall conception. For an extended interval within the twentieth century this element of Kummer's paintings turns out to were mostly forgotten, with the exception of a number of papers, between that are these via Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. within the mid 1950's, the idea of cyclotomic fields was once taken up back by way of Iwasawa and Leopoldt. Iwasawa considered cyclotomic fields as being analogues for quantity fields of the consistent box extensions of algebraic geometry, and wrote a very good series of papers investigating towers of cyclotomic fields, and extra in general, Galois extensions of quantity fields whose Galois crew is isomorphic to the additive staff of p-adic integers. Leopoldt targeting a hard and fast cyclotomic box, and tested a number of p-adic analogues of the classical advanced analytic classification quantity formulation. specifically, this led him to introduce, with Kubota, p-adic analogues of the advanced L-functions connected 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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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)*.

If x E ZeN) then x/N can be viewed be the smallest real number as an element of Q/Z. For any E R/Z we let ~ 0 in the residue class of t mod Z. What we want is for each positive integer k a polynomial PIc with rational coefficients, leading coefficient 1, such that the functions t

Character Sums we get l/IE(Q) = ex(CO(P»Jl(Cl(P»))lE:F'l = l/I(p)m/r. With a view towards (2), we conclude that (3) TI (1 -l/IE(Q)xmn(Q») = (1 Q\P = l/I(p)m/rxmn/ry TI = (1 - l/I(P)(xny 1 {mf'~ TI ,m=l (l - l/I(p)(eX)n). For this last step, we observe that the map gives a surjection of {lm -7- {lm/n and the inverse image of any element of {lm/r is a coset of {lr since r = (m, n). This makes the last step obvious. Substituting (3) in (2), we now find 1 + SE(XE, JlE)X m = = JI IJ TI (1 (1 - l/I(p\(ex)n(p» + Sex, Jl)eX) ~m~l This proves the theorem..

### Cyclotomic Fields by Dr. Serge Lang (auth.)

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