On the restricted Hilbert-Speiser and Leopoldt properties

Details On the restricted Hilbert-Speiser and Leopoldt properties On the restricted Hilbert-Speiser and Leopoldt properties

2009-05-17

arxiv.org/abs/0905.2737v2

Mathematics

Number Theory

15 pages, latex, to appear in Illinois Journal of Mathematics.

Scientific paper

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 O_L is free as an O_K[G]-module. If O_L is free over the associated order A_{L/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=C_p has prime order. We give examples with G=C_p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

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