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We prove that the Galois groups of the maximal unramified pro- 2 2 -extensions over the cyclotomic Z 2 Z_2 -extensions of certain real quadratic fields are metacyclic pro- 2 2 groups, and we give some criteria for the finiteness and examples relating to Greenberg’s conjecture.
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Yasushi Mizusawa (2010) studied this question.
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