LAUSR

Abstract Let k be a real quadratic number field with 2-class group C 2 ( k ) isomorphic to Z / 2 m Z x Z / 2 n Z , m ≥ 1 , n ≥ 2 , and let k 1 be the Hilbert 2-class field of k. We give complete criteria for C 2 ( k 1 ) to be cyclic when either d k , the discriminant of k, is divisible by only positive prime discriminants, or when the 2-class number of k 1 is greater than 2, and partial criteria for C 2 ( k 1 ) to be elementary cyclic when d k is divisible by a negative prime discriminant.