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$\mathbb Z_2$
-extensions of real quadratic fields
We give a family of real quadratic fields such that the 2-class field towers over their cyclotomic 
$\mathbb Z_2$
-extensions have metabelian Galois groups of abelian invariants 
$[2,2,2]$
. We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.
