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On the maximal unramified pro-2-extension over the cyclotomic 2 -extension of an imaginary quadratic field

Yasushi Mizusawa — 2010

Journal de Théorie des Nombres de Bordeaux

For the cyclotomic 2 -extension k of an imaginary quadratic field k , we consider the Galois group G ( k ) of the maximal unramified pro- 2 -extension over k . In this paper, we give some families of k for which G ( k ) is a metabelian pro- 2 -group with the explicit presentation, and determine the case that G ( k ) becomes a nonabelian metacyclic pro- 2 -group. We also calculate Iwasawa theoretically the Galois groups of 2 -class field towers of certain cyclotomic 2 -extensions.

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