The structure of the unramified abelian Iwasawa module of some number fields

For a given positive integer m, we determine an explicit infinite family of real quadratic number fields F, such that the unramified abelian Iwasawa module over the ‫ޚ‬ 2 -extension of F, is isomorphic to ‫)ޚ2/ޚ(‬ 2 m .• What about the structure of X ∞ (K )?• What is the least nonnegative integer n such that X ∞ (K ) ≃ A n (K )?