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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
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[6] Wintenberger, J.P., Structure Galois de limits proj d’unités locales, Comp. math., 42, 89-103, (1981)
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