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[1] LEPISTÖ, T.: On the growth of the first factor of the class number of the prime cyclotomic field, Ann. Acad. Sci. Fenn. Ser. A I, No. 577 (1974), 21 pp. [2] LOUBOUTIN, S.: Quelques formules exactes pour des moyennes de fonctions L de Dirichlet, Canad. Math. Bull. 36 (1993), 190–196. Addendum, Canad. Math. Bull. 37 (1994), p. 89. [3] LOUBOUTIN, S.: On the mean value of |L(1, χ)|2for odd primitive Dirichlet characters, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), 143–145. [4

Cyclotomic fields with unique factorization By /. Myron Masley* at Chicago and Hugh L. Montgomery* at Ann Arbor 1. Statement of results 2m For a natural number w>2, we let Cm = Q(em) be the ra-th cyclotornic field, of degree (m) over the field Q of rational numbers, we let hm denote the class number of Cm, and we let/? be a prime number. lfm is odd then Cm = C2m, so to avoid confusion we suppose always that m 2 (mod 4). Kummer conjectured that the integers in Cp have unique factorization (that is, hp= 1) if and only if p :g 19. This conjecture was in principle

:// [5] HORIE, K.: Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field, J. London Math. Soc. (2) 66 (2002), 257–275. [6] METSÄNKYLÄ, T.: Some divisibility results for the cyclotomic class number, Tatra Mt. Math. Publ. 11 (1997), 59–68. [7] STEVENHAGEN, P.: Class number parity for the pth cyclotomic field, Math. Comp. 63 (1994), 773–784. [8] WASHINGTON, L. C.: Class numbers and Z

p. Suppose p = l (mod 2μ) and denote the subfield of degree 2μ of 0(ζρ) with K. Then 2)(h(K(Q}. (This is again obtained by taking O = k, because q remains prime in Q(£p) and consequently in K.) Watabe, On class numbers ofsome cyclotomic fields 213 In particular, s 5 is a Fermat number and a primitive root mod 37 = 22 · 32 -i- 1, we have 2/fA(A:(C5)) where Kis the biquadratic subfield of O(C37). In order to obtain these theorems, we use the following three known propositions. (A proof of Prop. l is found in [3] p. 98, that of Prop. 2 in [1] or [2], and that of Prop

J. reine angew. Math. 544 (2002), 13—24 Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin New York 2002 Analytic ranks of elliptic curves over cyclotomic fields By Gautam Chinta at Providence 1. Introduction Let Lðs;EÞ be the L-function of an elliptic curve defined over Q, and let Lðs;E; wÞ be the L-function twisted by a primitive Dirichlet character w. It is of interest to know that many of the Lðs;E; wÞ are nonzero at the central point s ¼ 1=2 for w ranging over some set of primitive characters. In particular, letting Kq denote the

APPENDIX D p-adic Calculations in Cyclotomic Fields In this appendix we carry out some p-adic calculations in cyclotomic fields which are used in examples in Chapters 3 and 8. Everything here is basically well-known, due originally to Iwasawa and Coleman. For every n E z+ fix a primitive n-th root of unity (n such that (~n = (m for every m and n. By slight abuse of notation, for every n we will write Zp[JLnl = Z[JLn] ® Zp, the p-adic completion of Z(JLn], and similarly Qp(JLn) = Q(JLn) ® Qp. Define log : Zp(JLn][(Xj]X = Zp[JLn]x x (1 + XZp

On the integral basis of the maximal real subfleld of a cyclotomic field By Joseph J. Liang at Tampa Introduction Let C„ be a primitive n-th root of unity and #=(?(£,, + ζ'1) be the maximal real subfield of the n-th cyclotomic field Q(£n). It is proved in this paper that &»!_! {l, ς + C1, . . ., (ί,, + Ο 2 } is an integral basis of*. Throughout, the following notations will be used : n a positive integer greater than 2, Q the rational number field, ζη a primitive n-th root of unity, φ (n) the Euler φ-function of «, L = (C«) the «-th cyclotomic field, Κ=ζ)(ζη + ζ

Stickelberger's theorem for cyclotomic fields, in the spirit of Kummer and Thaine Lawrence C. Washington Abstract F. Thaine showed how to obtain annihilators of class groups of real abelian fields; we use Thaine's argument to give a simple proof of Stickelberger's theorem on annihilation of class groups of cyclotomic fields. In a recent paper, Francisco Thaine [3] showed how to obtain anni- hilators of class groups of real abelian fields. In the following, we show how Thaine's argument can be used to give a simple proof of Stickelberger's theorem on

On extensions of the maximal cyclotomic field having a given classical Galois group By G. V. Belyi at Vladimir Dedicated to L R. Shafaremch on the occasion of his 60. birthday § 1. Introduction In 1954 I. R. Safarevic had proved that there exist Galois extensions over an arbitrary algebraic number field having a given solvable Galois group. Nevertheless such extensions are not yet proved to exist for arbitrary groups. The recent success in the simple finite groups classification problem gives us some hope to solve the inverse Galois problem in the way similar to