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The commutativity of Galois groups of the maximal unramified pro-\(p\)-extensions over the cyclotomic \(\mathbb{Z}_{p}\)-extensions. II. (English) Zbl 1333.11101

Summary: Let \(p\) be an odd prime number and \(K_{\infty}\) the cyclotomic \(\mathbb{Z}_{p}\)-extension of a Galois \(p\)-extension \(K\) over an imaginary quadratic field. We consider the Galois group \(\tilde{X}(K_{\infty})\) of the maximal unramified pro-\(p\)-extension of \(K_{\infty}\). In this paper, under certain assumptions, we give certain \(K\) such that \(\tilde{X}(K_{\infty})\) is abelian. Also, we give an example such that a special value of the characteristic polynomial of the Iwasawa module of \(K_{\infty}\) determines whether \(\tilde{X}(K_{\infty})\) is abelian or not.
Part I see J. Number Theory 132, No. 4, 806–819 (2012; Zbl 1287.11125).

MSC:

11R23 Iwasawa theory
11R37 Class field theory

Citations:

Zbl 1287.11125
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Full Text: Euclid

References:

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