On unramified Galois 2-groups over ℤ₂-extensions of real quadratic fields

We prove that the Galois groups of the maximal unramified pro- 2 2 -extensions over the cyclotomic Z 2 Z₂ -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.