lnventiones math. 49, 87-97 (1978) In ve~ltioHes mathematicae ~ by Springer-Verlag 1978 The Non-p-Part of the Class Number in a Cyclotomic Zp-Extension Lawrence C. Washington Department of Mathematics, University of Maryland, College Park, Maryland, 20742, USA 1. Introduction Let k be an algebraic number field of finite degree over Q. A 71p-extension K/k is a Galois extension with Galois group isomorphic to the additive group Zp of p- adic integers. For any k we may obtain a 7/p-extension, called the cyclotomic 7/p- extension of k, by letting K be the appropriate subfield of ~ k(~p,) where ~m denotes a primitive m th root of unity. ">-1 For each integer n > 0 there is a unique subfield k, of K of degree p" over k. Let h, be the class number of k,. Iwasawa has shown that if pe, is the highest power of p dividing h, then there exist integers 2, p, v, independent of n, such that e, =p p"+,~ n + v for all sufficiently large n. Recently it was shown [1] that # =0 for cyclotomic 7/r-extensions of abelian number fields k (i.e. Gal(k/Q) is abelian). The purpose of this paper is to study the /-part of h,, where 1:4=p is prime. In fact, by extending the techniques of [-1], we prove the following result. Theorem. Let k be an abelian number field and K/k the cyclotomic Zp-extension of k. Let l+p be a prime and let I e" be the exact power of 1 dividing h n. Then e, is bounded independent of n (in fact e,, is constant for large n ). This result was conjectured in [23 by analogy with function fields over finite fields. It was proved there for abelian number fields k and the primes p = 2 and 3. Of course one conjectures that the above theorem remains valid for the cyclotomic 7Zp-extension of an arbitrary finite extension k of Q. However, there exist non-cyclotomic 2~p-extensions for which e, is unbounded [2]. It should be noted that the ideas of [1] show that the above theorem is effective; that is, it is possible to give computable bounds for when e, stops increasing. The proof of the theorem will be similar in many respects to the proof that =0, but there is one essential difference. Namely, in the study of the p-part of h, we were able to use certain power series constructed by Iwasawa, resulting from the isomorphism 7/p[[X]]___Zp[[~p]], the latter being the completion of the group ring. In the present case, we would have to work with ~l[[2~p]], which 0020-9910/78/0049/0087/$02.20