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 C-2(k) isomorphic to Z/2(m)Zx 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. (C) 2016 Elsevier Inc. All rights reserved.