Byott, Nigel P.

Carter, James E.

Greither, Cornelius

Johnston, Henri

Illinois Journal of Mathematics

Copyright © 2011 University of Illinois at Urbana-Champaign, Department of MathematicsLet 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 no

On the restricted Hilbert-Speiser and Leopoldt properties

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