Fix an odd prime number p and an abelian field K. Let U (resp. C) be the projective limit of the semi-local units at p (resp. of the cyclotomic units) of each intermediate field of the cyclotomic Z p -extension K ÂK. We study the Galois module structure of UÂC. We generalize results of Iwasawa and Gillard and sharpen a result of Greither. Academic Press

Abstract. For a prime number p and a number field k, let A∞ denote the projective limit of the p-parts of the ideal class groups of the intermediate fields of the cyclotomic Zp-extension over k. It is conjectured that A∞ is finite if k is totally real. When p is an odd prime and k is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where p divides the degree of k, we also obtain a rather simple criterion.

In the preceding papers, we studied the Iwasawa λ-invariant of the cyclotomic Z 2extension of Q( √ p ) for an odd prime number p using certain units and the invariants n (r) 0 and n 2 . In the present paper, we develop new criteria for Greenberg conjecture using n (r) 0 and n 2 .

In the preceding papers, two of authors developed criteria for Greenberg conjecture of the cyclotomic Z 2 -extension of k = Q( √ p ) with prime number p. Criteria and numerical algorithm in [5], [3] and [6] enable us to show λ 2 (k) = 0 for all p less than 10 5 except p = 13841, 67073. All the known criteria at present can not handle p = 13841, 67073. In this paper, we develop another criterion for λ 2 (k) = 0 using cyclotomic units and Iwasawa polynomials, which is considered a slight modification of the method of Ichimura and Sumida. Our new criterion fits the numerical examination and quickly shows that λ 2 (Q( √ p )) = 0 for p = 13841, 67073. So we announce here that λ 2 (Q( √ p )) = 0 for all prime numbers p less that 10 5 .

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