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The structure of the Iwasawa module associated with a $$\mathbb{Z}^ r_ p$$-extension of a $$p$$-adic local field of characteristic 0. (English) Zbl 0737.11029
Let $$K$$ be a finite extension of $$\mathbb{Q}_ p$$ of degree $$d$$ and assume $$L$$ is a Galois extension of $$K$$ such that $$\text{Gal}(L/K)\simeq\mathbb{Z}^ r_ p$$ for some $$r\geq 1$$. Let $$L^{ab,p}$$ be the maximal abelian pro-$$p$$-extension of $$L$$. Then $$M_ r=\text{Gal}(L^{ab,p}/L)$$ can be given the structure of a $$\Lambda_ r$$-module, where $$\Lambda_ r=\mathbb{Z}_ p[[T_ 1,\ldots,T_ r]]$$. The author proves that the $$\Lambda_ r$$- rank of $$M_ r$$ is $$d$$. When $$r=1$$ or 2, he shows that $$M_ r$$ is a free $$\Lambda_ r$$-module if $$L$$ contains no $$\ell$$-th roots of unity, and $$M_ r$$ has depth $$r$$ otherwise. When $$r\geq 3$$, he shows that the depth of $$M_ r$$ is 3, and obtains as a corollary that $$M_ r$$ is not free. This last corollary had previously been obtained by J. P. Wintenberger [Compos. Math. 42, 89-103 (1980; Zbl 0414.12008)] and by T. Nguyen-Quang-Do [J. Reine Angew. Math. 333, 133-143 (1982; Zbl 0481.12005)]. In the case $$r=1$$, the above results are due to K. Iwasawa [Ann. Math., II. Ser. 98, 246-326 (1973; Zbl 0285.12008)].
It has been pointed out to the reviewer that the results of the present paper also can be deduced from work of T. Nguyen-Quang-Do [Lect. Notes Math. 1068, 167-185 (1984; Zbl 0543.12007)], who uses a more cohomological approach than the present author.
##### MSC:
 11R23 Iwasawa theory 11S20 Galois theory
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##### References:
 [1] Greenberg, R., On the structure of certain Galois groups, Invent. math., 47, 85-99, (1978) · Zbl 0403.12004 [2] Hartshorne, R., (), GTM 52 [3] Iwasawa, K., On zi-extentions of algebraic numberfields, Annal. of math., 98, 246-326, (1973) [4] Nguyen-Quang-Do, Sur la structure Galois des corps locaux et la théorie d’Iwasawa, II, J. reine angew. math., 333, 133-143, (1982) · Zbl 0481.12005 [5] Serre, J.P., Algebra locale—multiplicities, () [6] Wintenberger, J.P., Structure Galois de limits proj d’unités locales, Comp. math., 42, 89-103, (1981)
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