Tame Galois module structure revisited

Abstract

In the last decades a lot of work has been done in the study of Hilbert-Speiser fields; in particular, criteria were developed which insure that a given field is not Hilbert-Speiser. The framework is Galois module structure of extensions of number fields. More precisely the object of our study is the property of an extension of number fields to have a normal integral basis: an extension L/K of number fields admits a NIB if O L is a rank 1 free O K [G]-module. The Hilbert-Speiser theorem states that for abelian number fields tameness is equivalent to the existence of a normal integral basis over Q, while in general we only have that an extension with NIB has to be tame. We will say that a number field K is Hilbert-Speiser if every tame abelian extension L/K has NIB, and C l -Hilbert-Speiser if every such extension with Galois group isomorphic to C l , the cyclic group of prime order l, has NIB.The first contribution to the topic of Hilbert-Speiser fields came from the important result contained in [GRRS99]: Q is the only Hilbert-Speiser field, i.e. for every number field K Q there exists a (cyclic of prime order) tame abelian extension that does not have NIB. Subsequent research went towards the finer problem of finding crieria for C l -Hilbert-Speiser fields. For instance, in [Car03], [Ich04] and [Yos09] there are some conditions for abelian C 2 and C 3 -Hilbert-Speiser fields, while from [Her05] and [GJ09] we know that in general C l -Hilbert-Speiser fields cannot be highly ramified if they are respectively totally imaginary or totally real.In this work, our intention is to consider a weakened version of NIB: we will say that a tame abelian extension L/K of number fields has a weak normal integral basis if M ⊗ OK[G] O L is free of rank 1 over M, where M is the maximal order of K [G]. WNIB's have been studied for instance in [Gre90], [Gre97] and [GJ12]. We shall ask the same questions as above substituting "WNIB" to "NIB" everywhere, and the condition that results from this substitution will be called "Leopoldt field" instead of "Hilbert-Speiser field". We are going to find mainly necessary conditions for number fields to be C l -Leopoldt, where as before l is a prime number, which also give criteria and (sometimes conditional) finiteness results for C l -Hilbert-Speiser fields; for instance we will see that this permits to correct an oversight contained in the article [Ich16], whose techniques, even though they were originally conceived to deal with Hilbert-Speiser fields, turn out to be supple enough to be applied to our problem as well.