Some real quadratic number fields with their Hilbert 2-class field having cyclic 2-class group

Let k be a real quadratic number field with 2-class group C2(k) isomorphic to Z/2mZxZ/2nZ, m≥1, n≥2, and let k1 be the Hilbert 2-class field of k. We give complete criteria for C2(k1) to be cyclic when either dk, the discriminant of k, is divisible by only positive prime discriminants, or when the 2-class number of k1 is greater than 2, and partial criteria for C2(k1) to be elementary cyclic when dk is divisible by a negative prime discriminant.