On the index of circular units in the full group of units of a compositum of quadratic fields

For a compositum of quadratic fields , where d₁,…,ds are square-free odd integers and , we study the group C of circular units of k. We construct a basis of C, compute the index of C in the full group of units of k and derive a lower bound for the divisibility of this index by a power of 2. These results give a lower bound for the divisibility of the class number of the maximal real subfield of k by a power of 2.