Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C-2 and C-3 denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type C-2: Q(rootm), where m is an element of {- 1, -3, -7}. Secondly, we give some necessary and sufficient conditions for a real quadratic field K = Q(rootm) to be a Hilbert-Speiser field of type C-2. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type C-3: Q(rootm), where m is an element of {- 11, -3, -2, 2, 5, 17, 41, 89}.