On the Equivalence of the Restricted Hilbert-speiser and Leopoldt Properties
نویسنده
چکیده
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 OKG-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 holds for many number fields K (in particular, for K/Q Galois) when G = Cp has prime order. In the process, we prove that if p ≥ 7 (or p = 5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type Cp.
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