proceedings of the american mathematical society Volume 61, Number 2, December 1976 THE CLASS NUMBER OF THE FIELD OF 5"TH ROOTS OF UNITY LAWRENCE C. WASHINGTON1 Abstract. Let h~ be the relative class number of the field of 5"th roots of unity. If / is any prime number, then the /-part of h~ is bounded indepen- dent of n. Let A:be a number field, K/k the cyclotomic Z^-extension of k, and kn the unique intermediate field of degree p" over k. In [4] it was conjectured that if / ^ p is any prime number then the /-part of the class number of kn is bounded independent of n. This conjecture arose from analogy with the case of function fields over finite fields, where Zp-extensions can be obtained by extending the field of constants. In this case it is not difficult to show that the /-part of the order of the group of divisor classes of degree zero is bounded independent of n [4]. For number fields, the conjecture has been proved when k/Q is abelian and p = 2 or 3. The main obstacle for larger primes p is the existence of p-adic (p — l)st roots of unity, which are, of course, harder to handle when p > 5. In this note we attack the case of the simplest Z5-extension, namely the one obtained by adjoining all 5"th roots of unity, for all n > 1, to the field of 5th roots of unity. Recall that for any imaginary abelian number field K there is a maximal real subfield K +, and since K/K+ is totally ramified (at oo) the class number h+ of K+ divides the class number h of K. The quotient h/h+ is called the relative class number h ~. Theorem. Let h~ be the relative class number of Q(f5n), where f5„ is a primitive 5"th root of unity. Let I be any prime number and let le- \\h~. Then e~ is bounded as n -^ oo. In fact, if m, is determined by 5m'||/4 — 1 and n > 2m, + 2, then l\h~/h~_v (la\\b means la\b, la+iJfb). Proof. Since 5 is a regular prime, 5\h~ for any n [2], Therefore, we assume / ¥= 5. The set of odd Dirichlet characters of Q(iV) is obtained as follows: Let Xp X2 be the odd characters for Q(f5)/Q and let xpn be a character of conductor 5" such that xpn(a) depends only on a4 mod 5" (therefore \\>n generates the Received by the editors March 1, 1976. AMS (MOS) subject classifications (1970). Primary 12A50, 12A35. 'Partially supported by NSF Grant MPS74-07491A01. © American Mathematical Society 1977 205 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use