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The commutativity of the Galois groups of the maximal unramified pro-\(p\)-extensions over the cyclotomic \(\mathbb Z_p\)-extensions. (English) Zbl 1287.11125

Summary: Let \(p\) be an odd prime number. For the cyclotomic \(\mathbb Z_p\)-extension \(F_{\infty }\) of a finite algebraic number field \(F\), we denote by \(\tilde L(F_\infty)\) the maximal unramified pro-\(p\)-extension of \(F_{\infty }\). In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for \(\text{Gal}(\tilde L(F_\infty)/F_\infty)\) to be abelian, and give sufficient conditions in certain special cases.

MSC:

11R23 Iwasawa theory
11R37 Class field theory
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