## 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

Zbl 1287.11125
Full Text:

### References:

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