On the restricted Hilbert-Speiser and Leopoldt properties
Byott, Nigel P.; Carter, James E.; Greither, Cornelius; et al.Johnston, Henri
Date: 1 January 2011
Article
Journal
Illinois Journal of Mathematics
Publisher
University of Illinois
Publisher DOI
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 ...
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
Mathematics and Statistics
Faculty of Environment, Science and Economy
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