On the restricted Hilbert-Speiser and Leopoldt properties

Abstract

Let G be a finite abelian group. A number field K is called a Hilbert–Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers OL is free as an OK[G]-module. If OL is free over the associated order AL/K for every G-Galois extension L/K, then K is called a Leopoldt field of type 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 in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G = Cp has prime order. We give examples with G = C5 to show that even the modified converse is false in general, and that the modified converse can hold when the original does not