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Let G be a finite abelian group. A number field K is called a Hilbert-Speiserfield of type G if, for every tame G-Galois extension L/K, the ring of integersO_L is free as an O_K[G]-module. If O_L is free over the associated orderA_{L/K} for every G-Galois extension L/K, then K is called a Leopoldt field oftype G. It is well-known (and easy to see) that if K is Leopoldt of type G,then K is Hilbert-Speiser of type G. We show that the converse does not hold ingeneral, but that a modified version does hold for many number fields K (inparticular, for K/Q Galois) when G=C_p has prime order. We give examples withG=C_p to show that even the modified converse is false in general, and that themodified converse can hold when the original does not.

On the restricted Hilbert-Speiser and Leopoldt properties