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 (m~(1/2)), where m ∈ {-1, -3, -7}. Secondly, we give some necessary and sufficient conditions for a real quadratic field K = Q (m~(1/2)) 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(m~(1/2)), where m ∈ {-11, -3, -2, 2, 5, 17, 41, 89}.
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